1911 Encyclopædia Britannica/Number

by George Ballard Mathews

NUMBER^[1] (through Fr. nombre, from Lat. numerus; from a root seen in Gr. νέμειν to distribute), a word generally expressive of quantity, the fundamental meaning of which leads on analysis to some of the most difficult problems of higher mathematics.

1. The most elementary process of thought involves a distinction within an identity-the A and the not-A within the sphere throughout which these terms are intelligible. Again A may be a generic quality found in different modes Aa, Ab, Ac, &c.; for instance, colour in the modes, red, green, blue and so on. Thus the notions of “one,” “two,” and the vague “many” are fundamental, and must have impressed themselves on the human mind at a very early, period: evidence of this is found in the grammatical distinction of singular, dual and plural which occurs in ancient languages of widely different races. A more definite idea of number seems to have been gradually acquired by realizing the equivalence, as regards plurality, of different concrete groups, such as the fingers of the right hand and those of the left. This led to the invention of a set of names which in the first instance did not suggest a numerical system, but denoted certain recognized forms of plurality, just as blue, red, green, &c., denote recognized forms of colour. Eventually the conception of the series of natural numbers became sufficiently clear to lead to a systematic terminology, and the science of arithmetic was thus rendered possible. But it is only in quite recent times that the notion of number has been submitted to a searching critical analysis: it is, in fact, one of the most characteristic results of modern mathematical research that the term number has been made at once more precise and more extensive.

2. Aggregates (also called manifolds or sets).—Let us assume the possibility of constructing or contemplating a permanent system of things such that (1) the system includes all objects to which a certain definite quality belongs; (2) no object without this quality belongs to the system; (3) each object of the system is permanently recognizable as the same thing, and as distinct from all other objects of the system. Such a collection is called an aggregate: the separate objects belonging to it are called its elements. An aggregate may consist of a single element.

It is further assumed that we can select, by a definite process, one or more elements of any aggregate ${\mbox{A}}$ at pleasure: these form another aggregate ${\mbox{B}}$. If any element of ${\mbox{A}}$ remains unselected, ${\mbox{B}}$ is said to be a part of ${\mbox{A}}$ (in symbols, ${\mbox{B}}<{\mbox{A}}$): if not, ${\mbox{B}}$ is identical with ${\mbox{A}}$. Every element of ${\mbox{A}}$ is a part of ${\mbox{A}}$. If ${\mbox{B}}<{\mbox{A}}$ and ${\mbox{C}}<{\mbox{B}}$, then ${\mbox{C}}<{\mbox{A}}$.

When a correspondence can be established between two aggregates ${\mbox{A}}$ and ${\mbox{B}}$ in such a way that to every element of ${\mbox{A}}$ corresponds one and only one element of ${\mbox{B}}$, and conversely, ${\mbox{A}}$ and ${\mbox{B}}$ are said to be equivalent, or to have the same power (or potency); in symbols, ${\mbox{A}}\sim {\mbox{B}}$. If ${\mbox{A}}\sim {\mbox{B}}$ and ${\mbox{B}}\sim {\mbox{C}}$, then ${\mbox{A}}\sim {\mbox{C}}$. It is possible for an aggregate to be equivalent to a part of itself: the aggregate is then said to be infinite. As an example, the aggregates $2,4,6,...2n$, &c., and $1,2,3,...n$, &c., are equivalent, but the first is only a part of the second.

3. Order.—Suppose that when any two elements $a,b$ of an aggregate ${\mbox{A}}$ are taken there can be established, by a definite criterion, one or other of two alternative relations, symbolized by $a<b$ and $a>b$, subject to the following conditions:-(1) If $a>b$, then $b<a$, and if $a<b$, then $b>a$; (2) If $a>b$ and $b>c$, then $a>c$. In this case the criterion is said to arrange the aggregate in order. An aggregate which can be arranged in order may be called ordinable. An ordinable aggregate may, in general, by the application of different criteria, be arranged in order in a variety of ways. According as $a<b$ or $a>b$ we shall speak of a as anterior or posterior to $b$. These terms are chosen merely for convenience, and must not be taken to imply any meaning except what is involved in the definitions of the signs $>$ and $<$ for the particular criterion in question. The consideration of a succession of events in time will help to show that the assumptions made are not self-contradictory. An aggregate arranged in order by a definite criterion will be called an ordered aggregate. Let $a,b$ be any two elements of an ordered aggregate, and suppose $a<b$. All the elements $c$ (if any) such that $a<c<b$ are said to fall within the interval $(a,b)$. If an element $b$, posterior to $a$, can be found so that no element falls within the interval $(a,b)$, then $a$ is said to be isolated from all subsequent elements, and $b$ is said to be the element next after $a$. So if $b'<a$, and no element falls within the interval $(b',a)$, then $a$ is isolated from all preceding elements, and $b'$ is the element next before $a$. As will be seen presently, for any assigned element $a$, either, neither, or both of these cases may occur.

An aggregate ${\mbox{A}}$ is said to be well-ordered (or normally ordered) when, in addition to being ordered, it has the following properties: (1) ${\mbox{A}}$ has a first or lowest element a which is anterior to all the rest; (2) if ${\mbox{B}}$ is any part of ${\mbox{A}}$, then ${\mbox{B}}$ has a first element. It follows from this that every part of a well-ordered aggregate is itself well-ordered. A well-ordered aggregate may or may not have a last element.

Two ordered aggregates ${\mbox{A}},{\mbox{B}}$ are said to be similar (${\mbox{A}}\simeq {\mbox{B}}$) when a one-one correspondence can be set up between their elements in such a way that if $b,b'$ are the elements of B which correspond to any two elements $a,a'$ of A, then $b>b'$ or $b<b'$ according as $a>a'$ or $a<a'$. For example, $(1,3,5,...)\simeq (2,4,6,...)$, because we can make the even number $2n$ correspond to the odd number $(2n-1)$ and conversely.

Similar ordered aggregates are said to have the same order-type. Any definite order-type is said to be the ordinal number of every aggregate arranged according to that type. This somewhat vague definition will become clearer as we proceed.

4. The Natural Scale.—Let $a$ be any element of a well-ordered aggregate ${\mbox{A}}$. Then all the elements posterior to $a$ form an aggregate ${\mbox{A}}'$, which is a part of ${\mbox{A}}$ and, by definition, has a first element $a'$. This element $a'$ is different from $a$, and immediately succeeds it in the order of ${\mbox{A}}$. (It may happen, of course, that $a'$ does not exist; in this case $a$ is the last element of ${\mbox{A}}$.) Thus in a well-ordered aggregate every element except the last (if there be a last element) is succeeded by a definite next element. The ingenuity of man has developed a symbolism by means of which every symbol is associated with a definite next succeeding symbol, and in this way we have a set of visible or audible signs 1, 2, 3, &c. (or their verbal equivalents), representing an aggregate in which (1) there is a definite order, (2) there is a first term, (3) each term has one next following, and consequently there is no last term. Counting a set of objects means associating them in order with the first and subsequent members of this conventional aggregate. The process of counting may lead to three different results: (1) the set of objects may be finite in number, so that they are associated with a part of the conventional aggregate which has a last term; (2) the set of objects may have the same power as the conventional aggregate; (3) the set of objects may have a higher power than the conventional aggregate. Examples of (2) and (3) will be found further on. The order-type of 1, 2, 3, &c., and of similar aggregates will be denoted by $\omega$; this is the first and simplest member of a set of transfinite ordinal numbers to be considered later on. Any finite number such as 3 is used ordinally as representing the order-type of 1, 2, 3 or any similar aggregate, and cardinally as representing the power of 1, 2, 3 or any equivalent aggregate. For reasons that will appear, $\omega$ is only used in an ordinal sense. The aggregate 1, 2, 3, &c., in any of its written or spoken forms, may be called the natural scale, and denoted by ${\mbox{N}}$. It has already been shown that ${\mbox{N}}$ is infinite: this appears in a more elementary way from the fact that $(1,2,3,4,...)\simeq (2,3,4,5,...)$, where each element of ${\mbox{N}}$ is made to correspond with the next following. Any aggregate which is equivalent to the natural scale or a part thereof is said to be countable.

5. Arithmetical Operations.—When the natural scale ${\mbox{N}}$ has once been obtained it is comparatively easy, although it requires a long process of induction, to define the arithmetical operations of addition, multiplication and involution, as applied to natural numbers. It can be proved that these operations are free from ambiguity and obey certain formal laws of commutation, &c., which will not be discussed here. Each of the three direct operations leads to an inverse problem which cannot be solved except under certain implied conditions. Let $a,b$ denote any two assigned natural numbers: then it is required to find natural numbers, $x,y,z$ such that

$a=b+x,\;a=by,\;a=z^{b}$

respectively. The solutions, when they exist, are perfectly definite, and may be denoted by $a-b,\;a/b$ and ${\sqrt[{b}]{a}}$; but they are only possible in the first case when $a>b$, in the second when $a$ is a multiple of $b$, and in the third when a is a perfect $b$th power. It is found to be possible, by the construction of certain elements, called respectively negative, fractional and irrational numbers, and zero, to remove all these restrictions.

6. There are certain properties, common to the aggregates with which we have next to deal, analogous to those possessed by the natural scale, and consequently justifying us in applying the term number to any one of their elements. They are stated here, once for all, to avoid repetition; the verification, in each case, will be, for the most part, left to the reader. Each of the aggregates in question (${\mbox{A}}$, suppose) is an ordered aggregate. If $\alpha ,\beta$ are any two elements of ${\mbox{A}}$, they may be combined by two definite operations, represented by $+$ and $\times$, so as to produce two definite elements of ${\mbox{A}}$ represented by $\alpha +\beta$ and $\alpha \times \beta$ (or $\alpha \beta$); these operations obey the formal laws satisfied by those of addition and multiplication. The aggregate ${\mbox{A}}$ contains one (and only one) element $\iota$, such that if $\alpha$ is any element of ${\mbox{A}}$ ($\iota$ included), then $\alpha +\iota >\alpha$, and $\alpha \iota >\alpha$. Thus ${\mbox{A}}$ contains the elements $\iota ,\iota +\iota ,\iota +\iota +\iota ,\&{\mbox{c}}.$, or, as we may write them, $\iota ,2\iota ,3\iota ,...m\iota ...$ such that $m\iota +n\iota =(m+n)\iota$ and $m\iota \times n\iota =mn\iota$; also $\iota <2\iota <3\iota ...$ We may express this by saying that ${\mbox{A}}$ contains an image of the natural scale. The element denoted by $\iota$ may be called the ground element of ${\mbox{A}}$.

7. Negative Numbers.—Let any two natural numbers $a,b$ be selected in a definite order $a,b$ (to be distinguished from $b,a$, in which the order is reversed). In this way we obtain from ${\mbox{N}}$ an aggregate of symbols $(a,b)$ which we shall call couples, от more precisely, if necessary, polar couples. This new aggregate may be arranged in order by means of the following rules:—

Two couples $(a,b),(a',b')$ are said to be equal if $a+b'=a'+b$. In other words $(a,b),(a',b')$ are then taken to be equivalent symbols for the same thing.

If $a+b'>a'+b$, we write $(a,b)>(a',b')$; and if $a+b'<a'+b$ we write $(a,b)<(a',b')$.

The rules for the addition and multiplication of couples are:

$(a,b)+(a',b')$	$=(a+a',b+b')$
$(a,b)\times (a',b')$	$=(aa'+bb',ab'+a'b).$

The aggregate thus defined will be denoted by ${\bar {\mbox{N}}}$; it may be called the scale of relative integers.

If $\iota$ denotes $(2,1)$ or any equivalent couple, $(a,b)+\iota =(a+2,b+1)>(a,b)$ and $(a,b)\times \iota =(2a+b,a+2b)=(a,b)$. Hence $\iota$ is the ground element of ${\bar {\mbox{N}}}$. By definition, $2\iota =\iota +\iota =(4,2)=(3,1)$: and hence by induction $m\iota =(m+1,1)$, where $m$ is any natural integer. Conversely every couple $(a,b)$ in which $a>b$ can be expressed by the symbol $(a-b)\iota$. In the same way, every couple $(a,b)$ in which $b>a$ can be expressed in the form $(b-a)\iota '$, where $\iota '=(1,2)$.

8. It follows as a formal consequence of the definitions that $\iota +\iota '=(2,1)+(1,2)=(3,3)=(1,1)$. It is convenient to denote $(1,1)$ and its equivalent symbols by $0$, because

$(a,b)+(1,1)=(a+1,b+1)$	$=(a,b)$
$(a,b)\times (1,1)=(a+b,a+b)$	$=(1,1)$;

hence $\iota +\iota '=0$, and we can represent ${\bar {\mbox{N}}}$ by the scheme—

$...3\iota ',2\iota ',\iota ',0,\iota ,2\iota ,3\iota ...$

in which each element is obtained from the next before it by the addition of $\iota$. With this notation the rules of operation may be written ($m,n$, denoting natural numbers)—

$m\iota +n\iota$	$=(m+n)\iota$	$m\iota '+n\iota '=(m+n)\iota '$
$m\iota +n\iota$	$=(m-n)\iota {\mbox{ if}}$	$m>n$
	$=(n-m)\iota '$	$m<n$

$m\iota \times n\iota =mn\iota ,\quad m\iota '\times n\iota '=mn\iota ,\quad m\iota \times n\iota '=mn\iota ',$

with the special rules for zero, that if $\alpha$ is any element of ${\bar {\mbox{N}}}$,

$\alpha +0=\alpha ,\quad \alpha \times 0=0$.

To each element, $\alpha$, of ${\bar {\mbox{N}}}$ corresponds a definite element $\alpha '$ such that $\alpha +\alpha '=0$; if $\alpha =0$, then $\alpha '=0$, but in every other case $\alpha ,\alpha '$ are different and may be denoted by $m\iota ,m\iota '$. The natural number $m$ is called the absolute value of $m\iota$ and $m\iota '$.

9. If $\alpha ,\beta$ are any two elements of ${\bar {\mbox{N}}}$, the equation $\xi +\beta =\alpha$ is satisfied by putting $\xi =\alpha +\beta '$. Thus the symbol $\alpha -\beta$ is always interpretable as $\alpha +\beta '$, and we may say that within ${\bar {\mbox{N}}}$ subtraction is always possible; it is easily proved to be also free from ambiguity. On the other hand, $\alpha /\beta$ is intelligible only if the absolute value of $\alpha$ is a multiple of the absolute value of $\beta$.

The aggregate ${\bar {\mbox{N}}}$ has no first element and no last element. At the same time it is countable, as we see, for instance, by associating the elements $0,a\iota ,b\iota '$ with the natural numbers $1,2a,2b+1$ respectively, thus—

$({\mbox{N}})\,1,2,3,4,5,6,\dots$
$({\bar {\mbox{N}}})\,0,\iota ,\iota ',2\iota ,2\iota ',3\iota \dots$

It is usual to write $+a$ (or simply $a$) for $a\iota$ and $-a$ for $a\iota '$; that this should be possible without leading to confusion or ambiguity is certainly remarkable.

10. Fractional Numbers.—We will now derive from ${\mbox{N}}$ a different aggregate of couples $[a,b]$ subject to the following rules:

The symbols $[a,b],[a',b'],$ are equivalent if $ab'=a'b$. According as $ab'$ is greater or less than $a'b$ we regard $[a,b]$ as being greater or less than $[a',b']$. The formulae for addition and multiplication are

$[a,b]+[a',b']$	$=[ab'+a'b,bb']$
$[a,b]\times [a',b']$	$=[aa',b']$.

All the couples $[a,a]$ are equivalent to $[1,1]$, and if we denote this by $\upsilon$ we have $[a,b]+\upsilon =[a+b,b]>[a,b],[a,b]\times \upsilon =[a,b]$, so that $\upsilon$ is the ground element of the new aggregate.

Again $2\upsilon =\upsilon +\upsilon =(2,1)$, and by induction $m\upsilon =[m,1]$. Moreover, if $a$ is a multiple of $b$, say $mb$, we may denote $[a,b]$ by $m\upsilon$.

11. The new aggregate of couples will be denoted by ${\mbox{R}}$. It differs from ${\mbox{N}}$ and ${\bar {\mbox{N}}}$ in one very important respect, namely, that when its elements are arranged in order of magnitude (that is to say, by the rule above given) they are not isolated from each other. In fact if $[a,b]=\alpha$, and $[a',b']=\alpha '$, the element $[a+a',b+b']$ lies between $\alpha$ and $\alpha '$; hence it follows that between any two different elements of ${\mbox{R}}$ we can find as many other elements as we please. This property is expressed by saying that ${\mbox{R}}$ is in close order when its elements are arranged in order of magnitude. Strange as it appears at first sight, ${\mbox{R}}$ is a countable aggregate; a theorem first proved by G. Cantor. To see this, observe that every element of R may be represented by a "reduced" couple $[a,b]$, in which $a,b$ are prime to each other. If $[a,b],[c,d]$ are any two reduced couples, we will agree that $[a,b]$ is anterior to $[c,d]$ if either (1) $a+b<c+d$, or (2) $a+b<c+d$ but $a<c$. This gives a new criterion by which all the elements of R can be arranged in the succession

$[1,1],[1,2],[2,1],[1,3],[3,1,],[1,4],[2,3],[3,2],[4.1]...$

which is similar to the natural scale.

The aggregate ${\mbox{R}}$, arranged in order of magnitude, agrees with ${\bar {\mbox{N}}}$ in having no least and no greatest element; for if $\alpha$ denotes any element $[a,b]$, then $[2a-1,2b]<\alpha$.

12. The division of one element of ${\mbox{R}}$ by another is always possible; for by definition

$[c,d]\times [ad,bc]=[acd,bcd]=[a,b]$,

and consequently $[a,b]\div [c,d]$ is always interpretable as $[ad,bc]$. As a particular case $[m,1]\div [n,1]=[m,n]$, so that every element of ${\mbox{R}}$ is expressible in one of the forms $m\upsilon ,m\upsilon /n\upsilon$. It is usual to omit the symbol $\upsilon$ altogether, and to represent the element $[m,n]$ by $m/n$, whether $m$ is a multiple of $n$ or not. Moreover, $m/1$ is written $m$, which may be done without confusion, because $m/1+n/1=(m+n)/1$, and $m/1\times n/1=mn/1$, by the rules given above.

13. Within the aggregate $\mathrm {R}$ subtraction is not always practicable; but this limitation may be removed by constructing an aggregate ${\bar {\mathrm {R} }}$ related to $\mathrm {R}$ in the same way as ${\bar {\mathrm {N} }}$ to $\mathrm {N}$. This may be done in two ways which lead to equivalent results. We may either form symbols of the type $(\alpha ,\beta )$ , where $\alpha ,\beta$ denote elements of $\mathrm {R}$, and apply the rules of § 7 ; or else form symbols of the type $[\alpha ,\beta ]$, where $\alpha ,\beta$ denote elements of ${\bar {\mathrm {N} }}$, and apply the rules of § 10. The final result is that ${\bar {\mathrm {R} }}$ contains a zero element, $0$, а ground element $\upsilon$, an element $\upsilon '$ such that $v+v'=0$, and a set of elements representable by the symbols ($m/n)\upsilon ,(m/n)\upsilon '$. In this notation the rules of operation are

${\frac {m}{n}}\upsilon +{\frac {m'}{n'}}\upsilon =\left({\frac {mn'+m'n}{nn'}}\right)\upsilon ,\quad {\frac {m}{n}}\upsilon '+{\frac {m'}{n'}}\upsilon '=\left({\frac {mn'+m'n}{nn'}}\right)\upsilon '$;
${\frac {m}{n}}\upsilon +{\frac {m'}{n'}}\upsilon '={\frac {mn'-m'n}{nn'}}\upsilon ,{\mbox{ or }}{\frac {m'n-mn'}{nn'}}\upsilon ',{\mbox{ as }}mn'>{\mbox{or}}<m'n;$;
${\frac {m}{n}}\upsilon \times {\frac {m'}{n'}}\upsilon ={\frac {mm'}{nn'}}\upsilon ={\frac {m}{n}}\upsilon '\times {\frac {m'}{n'}}\upsilon '\quad {\frac {m}{n}}\upsilon \times {\frac {m'}{n'}}\upsilon '={\frac {mm'}{nn'}}\upsilon '$
${\frac {m}{n}}\upsilon \div {\frac {m'}{n'}}\upsilon ={\frac {mn'}{m'n'}}\upsilon ={\frac {m}{n}}\upsilon '\div {\frac {m'}{n'}}\upsilon ',\quad {\frac {m}{n}}\upsilon \div {\frac {m'}{n'}}\upsilon '={\frac {mn'}{m'n'}}\upsilon '={\frac {m}{n}}\upsilon '\div {\frac {m'}{n'}}\upsilon ;$
$\alpha -\beta =\alpha +\beta ',{\mbox{ where }}\beta +\beta '=0$;
$\alpha +0=\alpha ,\ \alpha \times 0=0$.

Here $\alpha$ and $\beta$ denote any two elements of ${\bar {\mbox{R}}}$. If $\beta =(m/n)\upsilon$, then $\beta '=(m/n)\upsilon '$, and if $\beta =(m/n)\upsilon '$, then $\beta '=(m/n)\upsilon$. If $\beta =0$, then $\beta '=0$

14. When ${\bar {\mbox{R}}}$ is constructed by means of couples taken from ${\bar {\mbox{N}}}$, we must put $[m\iota ,n\iota ]=[m\iota ',n\iota ']=(m/n)\upsilon$, $[m\iota ,n\iota ']=[m\iota ',n\iota ]=(m/n)\upsilon '$, and $[0,\alpha ]=0$, if $\alpha$ is any element of ${\bar {\mbox{N}}}$ except $0$. The symbols $[0,0]$ and $[\alpha ,0]$ are inadmissible; the first because it satisfies the definition of equality (§ 10) with every symbol $[\alpha ,\beta ]$, and is therefore indeterminate; the second because, according to the rule of addition,

$[\alpha ,0]+[\iota ,\iota ]=[\alpha \iota ,0]=[\alpha ,0]$, which is inconsistent with $\xi +\upsilon >\xi$

In the same way, if $0$ denotes the zero element of ${\bar {\mbox{R}}}$, and $\xi$ any other element, the symbol $0/0$ is indeterminate, and $\xi /0$ inadmissible, because, by the formal rules of operation, $\xi /0+\upsilon =\xi /0$, which conflicts with the definition of the ground element $\upsilon$. It is usual to write $+{\frac {m}{n}}$ (or simply ${\frac {m}{n}}$) for ${\frac {m}{n}}\upsilon$, and $-{\frac {m}{n}}$ for ${\frac {m}{n}}\upsilon '$. Each of these elements is said to have the absolute value $m/n$. The criterion for arranging the elements of ${\bar {\mbox{R}}}$ in order of magnitude is that, if $\xi ,\eta$ are any two elements of it, $\xi >\eta$ when $\xi -\eta$ is positive; that is to say, when it can be expressed in the form $(m/n)\upsilon$.

15. The aggregate ${\bar {\mbox{R}}}$ is very important, because it is the simplest type of a field of rationality, or corpus. An algebraic corpus is an aggregate, such that its elements are representable by symbols $\alpha ,\beta$, &c., which can be combined according to the laws of ordinary algebra; every algebraic expression obtained by combining a finite number of symbols, by means of a finite chain of rational operations, being capable of interpretation as representing a definite element of the aggregate, with the single exception that division by zero is inadmissible. Since, by the laws of algebra, $\alpha -\alpha =0$, and $\alpha /\alpha =1$, every algebraic field contains ${\bar {\mbox{R}}}$, or, more properly, an aggregate which is an image of ${\bar {\mbox{R}}}$.

16. Irrational Numbers.—Let $\alpha$ denote any element of ${\bar {\mbox{R}}}$; then $\alpha$ and all lesser elements form an aggregate, ${\mbox{A}}$ say; the remaining elements form another aggregate ${\mbox{A}}'$, which we shall call complementary to ${\mbox{A}}$, and we may write ${\bar {\mbox{R}}}={\mbox{A}}+{\mbox{A}}'$. Now the essence of this separation of ${\bar {\mbox{R}}}$ into the parts ${\mbox{A}}$ and ${\mbox{A}}'$ may be expressed without any reference to $\alpha$ as follows:—

I. The aggregates ${\mbox{A}},{\mbox{A}}'$ are complementary; that is, their elements, taken together, make up the whole of ${\bar {\mbox{R}}}$.

II. Every element of ${\mbox{A}}$ is less than every element of ${\mbox{A}}'$.

III. The aggregate ${\mbox{A}}'$ has no least element. (This condition is artificial, but saves a distinction of cases in what follows.)

Every separation ${\bar {\mbox{R}}}={\mbox{A}}+{\mbox{A}}'$ which satisfies these conditions is called a cut (or section), and will be denoted by $({\mbox{A}},{\mbox{A}}')$. We have seen that every rational number $\alpha$ can be associated with a cut. Conversely, every cut $({\mbox{A}},{\mbox{A}})$ in which ${\mbox{A}}$ has a last element $\alpha$ is perfectly definite, and specifies $\alpha$ without ambiguity. But there are other cuts in which ${\mbox{A}}$ has no last element. For instance, all the elements ($\alpha$) of ${\bar {\mbox{R}}}$ such that either $\alpha \leq 0$, or $\alpha ^{2}<2$, form an aggregate ${\mbox{A}}$, while those for which $\alpha >0$ and $\alpha ^{2}>2$, form the complementary aggregate ${\mbox{A}}'$. This separation is a cut in which ${\mbox{A}}$ has no last element; because if $p/q$ is any positive element of ${\mbox{A}}$, the element $(3p+4q)/(2p+3q)$ exceeds $p/q$, and also belongs to ${\mbox{A}}$. Every cut of this kind is said to define an irrational number. The justification of this is contained in the following propositions:—

(1) A cut is a definite concept, and the assemblage of cuts is an aggregate according to definition; the generic quality of the aggregate being the separation of ${\bar {\mbox{R}}}$ into two complementary parts, without altering the order of its elements.

(2) The aggregate of cuts may be arranged in order by the rule that $({\mbox{A}},{\mbox{A}}')<({\mbox{B}},{\mbox{B}}')$ if ${\mbox{A}}$ is a part of ${\mbox{B}}$.

(3) This criterion of arrangement preserves the order of magnitude of all rational numbers.

(4) Cuts may be combined according to the laws of algebra, and, when the cuts so combined are all rational, the results are in agreement with those derived from the rational theory.

As a partial illustration of proposition (4) let $({\mbox{A}},{\mbox{A}}'),({\mbox{B}},{\mbox{B}}')$ be any two cuts ; and let ${\mbox{C}}'$ be the aggregate whose elements are obtained by forming all the values of $\alpha '+\beta '$, where $\alpha '$ is any element of ${\mbox{A}}'$ and $\beta '$ is any element of ${\mbox{B}}'$. Then if ${\mbox{C}}$ is the complement of ${\mbox{C}}'$, it can be proved that $({\mbox{C}},{\mbox{C}}')$ is a cut; this is said to be the sum of $({\mbox{A}},{\mbox{A}}')$ and $({\mbox{B}},{\mbox{B}}')$. The difference, product and quotient of two cuts may be defined in a similar way. If ${\mathfrak {n}}$ denotes the irrational cut chosen above for purposes of illustration, we shall have ${\mathfrak {n}}^{2}=({\mbox{C}},{\mbox{C}}')$ where ${\mbox{C}}'$ comprises all the numbers $\alpha '\beta '$ obtained by multiplying any two elements, $\alpha ',\beta '$ which are rational and positive, and such that $\alpha '^{2}>2,\beta '^{2}>2$. Since $\alpha '^{2}>2\beta '^{2}>2>4$ it follows that $\alpha '\beta '$ is positive and greater than $2$; it can be proved conversely that every rational number which is greater than $2$ can be expressed in the form $\alpha '\beta '$. Hence ${\mathfrak {n}}^{2}=2.$ so that the cut ${\mathfrak {n}}$ actually gives a real arithmetical meaning to the positive root of the equation $x^{2}=2$; in other words we may say that n defines the irrational number ${\sqrt {2}}$. The theory of cuts, in fact, provides a logical basis for the treatment of all finite numerical irrationalities, and enables us to justify all arithmetical operations involving the use of such quantities.

17. Since the aggregate of cuts (N say) has an order of magnitude, we may construct cuts in this aggregate. Thus if a is any element of N, and A is the aggregate which consists of a and all anterior elements of N, We may write N= A+A′, and (A, A′) is a cut in which A has a last element a. It is a remarkable fact that no other kind of cut in N is possible; in other words, every conceivable cut in N is defined by one of its own elements. This is expressed by saying that N is a continuous aggregate, and N itself is referred to as the numerical continuum of real numbers. The property of continuity must be carefully distinguished from that of close order (§ 11); a continuous aggregate is necessarily in close order, but the converse is not always true. The aggregate N is not countable.

18. Another way of treating irrationals is by means of sequences. A sequence is an unlimited succession of rational numbers

a₁, a₂, a₃ . . . a_m, a_m+1 . . .

(in order-type 𝜔) the elements of which can be assigned by a definite rule, such that when any rational number 𝜖, however small, has been fixed, it is possible to find an integer m, so that for all positive integral values of n the absolute value of (a_m+n − a_m) is less than 𝜖. Under these conditions the sequence may be taken to represent a definite number, which is, in fact, the limit of a_m when m increases without limit. Every rational number a can be expressed as a sequence in the form (a, a, a, . . .), but this is only one of an infinite variety of such representations, for instance—

1 ＝ (.9, .99, .999, . . .) ＝ $\left({1 \over 2},{3 \over 4},{7 \over 8},...{2^{n}-1 \over 2^{n}}...\right)$

and so on. The essential thing is that we have a mode of representation which can be applied to rational and irrational numbers alike, and provides a very convenient symbolism to express the results of arithmetical operations. Thus the rules for the sum and product of two sequences are given by the formulae

$(a_{1},\,a_{2},\,a_{3},\,.\,.\,.)+(b_{1},\,b_{2},\,b_{3})=(a_{1}+b_{1},\,a_{2}+b_{2},\,a_{3}+b_{3}\,.\,.\,.)$

$(a_{1},\,a_{2},\,a_{3},\,.\,.\,.)\times (b_{1},\,b_{2},\,b_{3},\,.\,.\,.)=(a_{1}b_{1},\,a_{2}b_{2},\,a_{3}b_{3}\,.\,.\,.)$

from which the rules for subtraction and division may be at once inferred. It has been proved that the method of sequences is ultimately equivalent to that of cuts. The advantage of the former lies in its convenient notation, that of the latter in giving a clear definition of an irrational number without having recourse to the notion of a limit.

19. Complex Numbers.—If 𝛼 is an assigned number, rational or irrational, and n a natural number, it can be proved that there is a real number satisfying the equation $x^{n}=a$, except when n is even and a is negative: in this case the equation is not satisfied by any real number whatever. To remove the difficulty we construct an aggregate of polar couples {x, y}, where x, y are any two real numbers, and define the addition and multiplication of such couples by the rules

$\{x,\ y\}+\{x',\ y'\}=\{x+x',\ y+y'\}$

$\{x,\ y\}\times \{x',\ y'\}=\{xx'-yy',xy'-x'y\}$

We also agree that {x, y} < {x′, y′}, if x<x′ or if x=x′ and y<y′. It follows that the aggregate has the ground element {1, 0}, which we may denote by 𝜎; and that, if we write 𝜏 for the element {0, 1},

𝜏²={–1, 0} = –𝜎.

Whenever m, n are rational, {m, n} = m𝜎+n𝜏, and we are thus justified in writing, if we like, x𝜎+y𝜏 for {x, y} in all circumstances. A further simplification is gained by writing x instead of x𝜎, and regarding 𝜏 as a symbol which is such that 𝜏²= −1, but in other respects obeys the ordinary laws of operation. It is usual to write i instead of τ; we thus have an aggregate J of complex numbers x+yi. In this aggregate, which includes the real continuum as part of itself, not only the four rational operations (excluding division by {0, 0}, the zero element), but also the extraction of roots, may be effected without any restriction. Moreover (as first proved by Gauss and Cauchy), if a₀, a₁ . . . a_n an are any assigned real or complex numbers, the equation

$a_{0}z^{n}+a_{1}z^{n-1}+...+a_{n-1}z+a_{n}=0$,

is always satisfied by precisely n real or complex values of z, with a proper convention as to multiple roots. Thus any algebraic function of any finite number of elements of J is also contained in J, which is, in this sense, a closed arithmetical field, just as N is when we restrict ourselves to rational operations. The power of J is the same as that of N.

20. Transfinite Numbers.—The theory of these numbers is quite recent, and mainly due to G. Cantor. The simplest of them, 𝜔, has been already defined (§ 4) as the order-type of the natural scale. Now there is no logical difficulty in constructing a scheme

u₁, u₂, u₃ . . . | v₁,

indicating a well-ordered aggregate of type 𝜔 immediately followed by a distinct element v₁: for example, we may think of all positive odd integers arranged in ascending order of magnitude and then think of the even number 2. A scheme of this kind is said to be of order-type (𝜔+1); and it will be convenient to speak of (𝜔+1) as the index of the scheme. Similarly we may form arrangements corresponding to the indices

𝜔+2, 𝜔+3 . . . 𝜔+n,

where n is any positive integer. The scheme

u₁, u₂, u₃ . . . | v₁, v₂, v₃ . . .

is associated with 𝜔+𝜔 = 2𝜔;

u₁₁, u₁₂, u₁₃ . . . | u₂₁, u₂₂, u₂₃ . . . | . . . | u_n1, u_n2, . . . | . . .

with 𝜔.𝜔 or 𝜔²; and so on. Thus we may construct arrangements of aggregates corresponding to any index of the form

𝜙(𝜔)＝a𝜔ⁿ+bωⁿ⁻¹+ . . . +k𝜔+l,

where n, a, b, . . . l are all positive integers.

We are thus led to the construction of a scheme of symbols—

${\begin{matrix}I.&\ 1,\ 2,\ 3,\ .\ .\ .\ n\ .\ .\ .\ \qquad \qquad \qquad \qquad \quad \\II.&{\begin{cases}\omega ,\ \omega +1,\ .\ .\ .\ \omega +n,\ .\ .\ .\ \\2\omega ,\ 2\omega +1,\ .\ .\ .\ 2\omega +n,\ .\ .\ .\ \\\vdots \\\omega ^{2},\ \omega ^{2}+1,\ \omega ^{2}+2,\ .\ .\ .\ \omega ^{2}+n,\ .\ .\ .\ \\\vdots \\\phi (\omega ),\ \phi (\omega )+1,\ .\ .\ .\ \phi (\omega )+n,\ .\ .\ .\ \\\vdots \end{cases}}\\III.&{\begin{cases}\omega ^{\omega },\ \omega ^{\omega }+1,\ .\ .\ .\ \omega ^{\omega }+n,\ .\ .\ .\ \\\vdots \\\omega ^{\phi (\omega )},\ \omega ^{\phi (\omega )}+1,\ .\ .\ .\ \omega ^{\phi (\omega )}+n,\ .\ .\ .\ \\\vdots \end{cases}}\end{matrix}}$

The symbols 𝜙(𝜔) form a countable aggregate: so that we may, if we like (and in various ways), arrange the rows of block (II.) in a scheme of type 𝜔: we thus have each element 𝛼 succeeded in its row by (𝛼+1), and the row containing 𝜙(𝜔) succeeded by a definite next row. The same process may be applied to (III.), and we can form additional blocks (IV.), (V.), &c., with first elements $\omega _{4}=\omega ^{\omega ^{\omega },}$ $\omega _{5}=\omega ^{\omega _{4},}$ &c. All the symbols in which 𝜔 occurs are called transfinite ordinal numbers.

21. The index of a finite set is a definite integer however the set may be arranged; we may take this index as also denoting the power of the set, and call it the number of things in the set. But the index of an infinite ordinable set depends upon the way in which its elements are arranged; for instance, ind. $(1,\ 2,\ 3,\ .\ .\ .\ )=\omega$, but ind. $(1,\ 3,\ 5,\ .\ .\ .\ |\ 2,\ 4,\ 6,\ .\ .\ .\ )=2\omega$. Or, to take another example, the scheme—

${\begin{matrix}1,\ 3,\ 5,\ .\ .\ .\ (2n-1)\ .\ .\ .\ \quad \\2,\ 6,\ 10,\ .\ .\ .\ 2\ (2n-1)\ .\ .\ .\ \\\vdots \ \ \ \vdots \ \ \ \ \ \vdots \qquad \qquad \qquad \qquad \quad \ \\\quad \qquad \qquad 2^{m},\ 2^{m}\ .\ 3,\ 2^{m}\ .\ 5,\ .\ .\ .\ 2^{m}\ (2n-1)\ .\ .\ .\ \\\vdots \qquad \qquad \qquad \qquad \qquad \qquad \end{matrix}}$

where each row is supposed to follow the one above it, gives a permutation of (1, 2, 3, . . . ), by which its index is changed from 𝜔 to 𝜔². It has been proved that there is a permutation of the natural scale, of which the index is 𝜙(𝜔), any assigned element of (II.); and that, if the index of any ordered aggregate is 𝜙(𝜔), the aggregate is countable. Thus the power of all aggregates which can be associated with indices of the class (II.) is the same as that of the natural scale; this power may be denoted by a. Since a is associated with all aggregates of a particular power, independently of the arrangement of their elements, it is analogous to the integers, 1, 2, 3, &c, when used to denote powers of finite aggregates; for this reason it is called the least transfinite cardinal number.

22. There are aggregates which have a power greater than a: for instance, the arithmetical continuum of positive real numbers, the power of which is denoted by c. Another one is the aggregate of all those order-types which (like those in II. above) are the indices of aggregates of power a. The power of this aggregate is denoted by ^א₁. According to Cantor's theory it is the transfinite cardinal number next superior to a, which for the sake of uniformity is also denoted by ^א₀. It has been conjectured that ^א₁＝c, but this has neither been verified nor disproved The discussion of the aleph-numbers is still in a controversial stage (November 1907) and the points in debate cannot be entered upon here.

23. Transfinite numbers, both ordinal and cardinal, may be combined by operations which are so far analogous to those of ordinary arithmetic that it is convenient to denote them by the same symbols. But the laws of operation are not entirely the same; for instance, 2𝜔 and 𝜔2 have different meanings: the first has been explained, the second is the index of the scheme (a₁b₁ | a₂b₂ | a₃b₃ | . . . | a_nb_n | . . . ) or any similar arrangement. Again if n is any positive integer, na＝aⁿ＝a. It should also be observed that according to Cantor’s principles of construction every ordinal number is succeeded by a definite next one; but that there are definite ordinal numbers (e.g. 𝜔, 𝜔²) which have no ordinal immediately preceding them.

24. Theory of Numbers.—The theory of numbers is that branch of mathematics which deals with the properties of the natural numbers. As Dirichlet observed long ago, the whole of the subject would be coextensive with mathematical analysis in general; but it is convenient to restrict it to certain fields where the appropriateness of the above definition is fairly obvious. Even so, the domain of the subject is becoming more and more comprehensive, as the methods of analysis become more systematic and more exact.

The first noteworthy classification of the natural numbers is into those which are prime and those which are composite. A prime number is one which is not exactly divisible by any number except itself and 1; all others are composite. The number of primes is infinite (Eucl. Elem. ix. 20), and consequently, if n is an assigned number, however large, there is an infinite number (a) of primes greater than n.

If m, n are any two numbers, and m>n, we can always find a definite chain of positive integers (q₁, r₁), (q₂, r₂), &c., such that

m＝q₁n+r₁, n＝q₂r₁+r₂, r₁＝q₃r₂+r₃, &c.

with n>r₁>r₂>r₃ . . .; the process by which they are calculated will be called residuation. Since there is only a finite number of positive integers less than n, the process must terminate with two equalities of the form

r_h−2＝q_hr_h−1+r_h,  r_h−1＝q_h+1r_h.

Hence we infer successively that r_h is a divisor of r_h−1, r_h−2, . . . r₁, and finally of m and n. Also r_h is the greatest common factor of m, n: because any common factor must divide r₁, r₂, and so on down to r_h: and the highest factor of r_h is r_h itself. It will be convenient to write r_h=dv (m, n). If r_h = 1, the numbers m, n are said to be prime to each other, or co-primes.

25. The foregoing theorem of residuation is of the greatest importance; with the help of it we can prove three other fundamental propositions, namely:—

(1) If m, n are any two natural numbers, we can always find two other natural numbers x, y such that

dv(m,n)＝xm−yn.

(2) If m, n are prime to each other, and p is a prime factor of mn, then p must be a factor of either m or n.

(3) Every number may be uniquely expressed as a product of prime factors.

Hence if n＝p ^𝛼q ^𝛽r ^𝛾 . . . is the representation of any number n as the product of powers of different primes, the divisors of n are the terms of the product (1+p+p ²+ . . . +p ^𝛼) (1+q+ . . . +q ^𝛽) (1+r . . . +r ^𝛾) their number is (𝛼+1) (𝛽+1) (𝛾+1) . . .; and their sum is ∏(p ^𝛼+1−1)÷∏(p−1). This includes 1 and n among the divisors of n.

26. Totients.—By the totient of n, which is denoted, after Euler, by 𝜙(n), we mean the number of integers prime to n, and not exceeding n. If n=p ^𝛼, the numbers not exceeding n and not prime to it are p, 2p, . . . (p ^𝛼−p), p ^𝛼 of which the number is p ^𝛼−1: hence 𝜙(p ^𝛼)=p ^𝛼−p ^𝛼−1. If m, n are prime to each other, 𝜙(mn)=𝜙(m)𝜙(n); and hence for the general case, if n=p ^𝛼q ^𝛽r ^𝛾 . . . ,𝜙(n)=∏p ^𝛼−1(p−1), where the product applies to all the different prime factors of n. If d₁, d₂, &c., are the different divisors of n,

𝜙(d₁)+𝜙(d₂)+ . . . ＝n.

For example 15=𝜙(15)+𝜙(5)+𝜙(3)+𝜙(1)=8+4+2+1.

27. Residues and congruences.—It will now be convenient to include in the term “number” both zero and negative integers. Two numbers a, b are said to be congruent with respect to the modulus m, when (a−b) is divisible by m. This is expressed by the notation a≡b (mod m), which was invented by Gauss. The fundamental theorems relating to congruences are

If	a≡b and c≡d (mod m), then a±c≡b±d, and ab≡cd.
If	ha≡hb(mod m) then a≡b (mod m/d), where d＝dv(h, m).

Thus the theory of congruences is very nearly, but not quite, similar to that of algebraic equations. With respect to a given modulus m the scale of relative integers may be distributed into m classes, any two elements of each class being congruent with respect to m. Among these will be 𝜙(m) classes containing numbers prime to m. By taking any one number from each class we obtain a complete system of residues to the modulus m. Supposing (as we shall always do) that m is positive, the numbers 0, 1, 2, . . . (m-1) form a system of least positive residues; according as m is odd or even, 0, ±1, ±2, . . . ±1/2 (m−1), or 0, ±1, ±2, . . . ±1/2(m−2),1/2m form a system of absolutely least residues.

28. The Theorems of Fermat and Wilson.—Let r₁, r₂, . . . r_t where t=𝜙(m), be a complete set of residues prime to the modulus m. Then if x is any number prime to m, the residues xr₁, xr₂, . . . xr_t also form a complete set prime to m (§ 27). Consequently xr₁·xr₂ . . . xr_t≡r₁r₂ . . .r_t, and dividing by r₁r₂ . . . r_t, which is prime to the modulus, we infer that

x^𝜙(^m)≡1(mod m).

which is the general statement of Fermat’s theorem. If m is a prime p, it becomes x^p−1≡1 (mod p). For a prime modulus p there will be among the set x, 2x, 3x, . . . (p−1)x just one and no more that is congruent to 1: let this be xy. If y≡x, we must have x²−1＝(x−1) (x+1)≡0, and hence x≡±1: consequently the residues 2, 3, 4, . . . (p−2) can be arranged in 1/2 (p−3) pairs (x, y) such that xy≡1. Multiplying them all together, we conclude that 2.3.4. . . .(p−2)≡1 and hence, since 1.(p−1)≡−1,

(p−1)!≡−1 (mod p).

which is Wilson’s theorem. It may be generalized, like that of Fermat, but the result is not very interesting. If m is composite (m−1)!+1 cannot be a multiple of m: because m will have a prime factor p which is less than m, so that (m−1)!≡0 (mod p). Hence Wilson’s theorem is invertible: but it does not supply any practical test to decide whether a given number is prime.

29. Exponents, Primitive Roots, Indices.—Let p denote an odd prime, and x any number prime to p. Among the powers x, x ², x ³, . . . x ^p−1 there is certainly one, namely x ^p−1, which ≡1 (mod p); let x ^e be the lowest power of x such that x ^e≡1. Then e is said to be the exponent to which x appertains (mod p): it is always a factor of (p−1) and can only be 1 when x≡1. The residues x for which e=p−1 are said to be primitive roots of p. They always exist, their number is 𝜙(p−1), and they can be found by a methodical, though tedious, process of exhaustion. If g is any one of them, the complete set may be represented by g, g ^a, g ^b, . . . &c. where a, b, &c., are the numbers less than (p−1) and prime to it, other than 1. Every number x which is prime to p is congruent, mod p, to gⁱ, where i is one of the numbers 1, 2, 3, . . . (p−1); this number i is called the index of x to the base g. Indices are analogous to logarithms: thus

${\mbox{ind}}_{g}(xy)\equiv {\mbox{ind}}_{g}x+{\mbox{ind}}_{g}y,\ {\mbox{ind}}_{g}(x^{h})\equiv h\ {\mbox{ind}}_{g}x\ ({\mbox{mod}}\ {\overline {p-1}})$

Consequently tables of primitive roots and indices for different primes are of great value for arithmetical purposes. Jacobi’s Canon Arithmeticus gives a primitive root, and a table of numbers and indices for all primes less than 1000.

For moduli of the forms 2p, p^m, 2p^m there is an analogous theory (and also for 2 and 4); but for a composite modulus of other forms there are no primitive roots, and the nearest analogy is the representation of prime residues in the form 𝛼^x 𝛽^y 𝜒^z . . . , where 𝛼, 𝛽, 𝛾 are selected prime residues, and x, y, z, . . . are indices of restricted range. For instance, all residues prime to 48 can be exhibited in the form 5^x 7^y 13^z, where x=0, 1, 2, 3; y=0, 1; z=0, 1; the total number of distinct residues being 4.2.2=16=𝜙(48), as it should be.

30. Linear Congruences.—The congruence a′x≡b′ (mod m′) has no solution unless dv(a', m') is a factor of b′. If this condition is satisfied, we may replace the given congruence by the equivalent one ax≡b (mod m), where a is prime to b as well as to m. By residuation (§§ 24, 25) we can find integers h, k such that ah−mk=1, and thence obtain x≡bh (mod m) as the complete solution of the given congruence. To the modulus m′ there are m′/m incongruent solutions. For example, 12x≡30 (mod 21) reduces to 2x≡5 (mod 7) whence x≡6 (mod 7)≡6, 13, 20 (mod 21). There is a theory of simultaneous linear congruences in any number of variables, first developed with precision by Smith. In any particular case, it is best to replace as many as possible of the given congruences by an equivalent set obtained by successively eliminating the variables x, y, z, . . . in order. An important problem is to find a number which has given residues with respect to a given set of moduli. When possible, the solution is of the form $x\equiv a\ ({\mbox{mod}}\ m)$, where m is the least common multiple of the moduli. Supposing that p is a prime, and that we have a corresponding table of indices, the solution of $ax\equiv b\ ({\mbox{mod}}\ p)$ can be found by observing that ${\mbox{ind}}\ x\equiv {\mbox{ind}}\ b-{\mbox{ind}}\ a\ ({\mbox{mod}}\ {\overline {p-1}})$.

31. Quadratic Residues. Law of Reciprocity.—To an odd prime modulus p, the numbers 1, 4, 9, . . . (p−1)² are congruent to 1/2(p−1) residues only, because $(p-x)^{2}=x^{2}$. Thus for $p=5$, we have $1,4,9,16\equiv 1,4,4,1$ respectively. There are therefore 1/2(p−1) quadratic residues and 1/2(p−1) quadratic non-residues prime to p; and there is a corresponding division of in congruent classes of integers with respect to p. The product of two residues or of two non residues is a residue; that of a residue and a non-residue is a non residue; and taking any primitive root as base the index of any number is even or odd according as the number is a residue or a non residue. Gauss writes aRp, aNp to denote that a, is a residue or non residue of p respectively.

Given a table of indices, the solution of $x^{2}\equiv a\ ({\mbox{mod}}\ p)$ when possible, is found from 2ind x≡ind a (mod p−1), and the result may be written in the form $x\equiv \pm r\ ({\mbox{mod}}\ p)$. But it is important to discuss the congruence $x^{2}\equiv a$ without assuming that we have a table of indices. It is sufficient to consider the case $x^{2}\equiv q\ ({\mbox{mod}}\ p)$, where q is a positive prime less than p; and the question arises whether the quadratic character of q with respect to p can be deduced from that of p with respect to q. The answer is contained in the following theorem, which is called the law of quadratic reciprocity (for real positive odd primes): if p, q are each or one of them of the form 4n+1, then p, q are each of them a residue, or each a non-residue of the other; but if p, q are each of the form 4n+3, then according as p is a residue or non-residue of q we have q a non-residue or a residue of p.

Legendre introduced a symbol $\left({\frac {m}{q}}\right)$ which denotes + 1 or −1 according as mRq or mNq being a positive odd prime and m any number prime to q); with its help we may express the law of reciprocity in the form

$\left({p \over q}\right)\left({q \over p}\right)=(-1)^{{\frac {1}{4}}(p-1)(q-1)}$

This theorem was first stated by Legendre, who only partly proved it; the first complete proof, by induction, was published by Gauss, who also discovered five (or six) other more or less independent proofs of it. Many others have since been invented.

There are two supplementary theorems relating to −1 and 2 respectively, which, may be expressed in the form

$\left({-1 \over p}\right)=(-1)^{{1 \over 2}(p-1)}$, $\left({2 \over p}\right)=(-1)^{{1 \over 8}(p^{2}-1)}$

where p is any positive odd prime.

It follows from the definition that

$\left({p_{1}^{\alpha }p_{2}^{\beta }p_{3}^{\gamma }... \over q}\right)=\left({p_{1} \over q}\right)^{\alpha }\left({p_{2} \over q}\right)^{\beta }\left({p_{3} \over q}\right)^{\gamma }...$

and that $\left({m \over q}\right)=\left({m' \over q}\right)$ if $m=m'\ ({\mbox{mod}}\ q)$. As a simple application of the law of reciprocity, let it be required to find the quadratic character of 11 with respect to 1907. We have

$\left({11 \over 1907}\right)=-\left({1907 \over 11}\right)=-\left({6 \over 11}\right)=1$

because 6N11. Hence 11R1907.

Legendre's symbol was extended by Jacobi in the following manner. Let P be any positive odd number, and let p, p', p", &c. be its (equal or unequal) prime factors, so that P=pp'p" .... Then if Q is any number prime to P, we have a generalized symbol defined by

$\left({{\mbox{Q}} \over {\mbox{P}}}\right)=\left({{\mbox{Q}} \over p}\right)\left({{\mbox{Q}} \over p'}\right)\left({{\mbox{Q}} \over p''}\right)...$

This symbol obeys the law that, if Q is odd and positive,

$\left({{\mbox{P}} \over {\mbox{Q}}}\right)\left({{\mbox{Q}} \over {\mbox{P}}}\right)=(-1)^{{1 \over 4}\left({\mbox{P}}-1\right)\left({\mbox{Q}}-1\right)}$

with the supplementary laws

$\left({-1 \over {\mbox{P}}}\right)=(-1)^{{1 \over 2}\left({\mbox{P}}-1\right)}$, $\left({2 \over {\mbox{P}}}\right)=(-1)^{{1 \over 8}\left({\mbox{P}}^{2}-1\right)}$

It is found convenient to add the conventions that

$\left({{\mbox{Q}} \over -{\mbox{P}}}\right)=\left({{\mbox{Q}} \over {\mbox{P}}}\right)$

when Q and P are both odd; and that the value of the symbol is 0 when P, Q are not co-primes.

In order that the congruence $x^{2}\equiv a\ ({\mbox{mod}}\ m)$ may have a solution it is necessary and sufficient that a be a residue of each distinct prime factor of m    If these conditions are all satisfied, and $m=2^{\kappa }p^{\lambda }q^{\mu }$, where p, q, &c., are the distinct odd prime factors of m, being t in all, the number of incongruent solutions of the given congruence is $2^{t}$, $2^{t+1}$ or $2^{t+2}$, according as $\kappa <2$, $\kappa =2$, or $\kappa >2$ respectively. The actual solutions are best found by a process of exhaustion. It should be observed that $\left({a \over m}\right)=1$ is a necessary but not a sufficient condition for the possibility of the congruence.

32. Quadratic forms.—It will be observed that the solution of the linear congruence $ax\equiv b\ ({\mbox{mod}}\ m)$ leads to all the representations of b in the form $ax+my$, where x, y are integers. Many of the earliest researches in the theory of numbers deal with particular cases of the problem: given four numbers m, a, b, c, it is required to find all the integers x, y (if there be any) which satisfy the equation $ax^{2}+bxy+cy^{2}=m$. Fermat, for instance, discovered that every positive prime of the form 4n+1 is uniquely expressible as the sum of two squares. There is a corresponding arithmetical theory for forms of any degree and any number of variables; only those of linear forms and binary quadratics are in any sense complete, as the difficulty of the problem increases very rapidly with the increase of the degree of the form considered or of the number of variables contained in it.

The form $ax^{2}+bxy+cy^{2}$ will be denoted by $(a,\ b,\ c\ )(x,\ y)^{2}$ or more simply by $(a,\ b,\ c\ )$ when there is no need of specifying the variables. If k is the greatest common factor of a, b, c,    we may write $(a,\ b,\ c)=k(a',\ b',\ c')$ where $(a',\ b',\ c')$ is a primitive form, that is, one for which ${\mbox{dv}}\ (a',\ b',\ c')=1$. The other form is than said to be derived from $(a',\ b',\ c')$ and to have a divisor k. For the present we shall concern ourselves only with primitive forms. Writing ${\mbox{D}}=b^{2}-4ac$, the invariant D is called the determinant of $(a,b,c)$, and there is a first classification of forms into definite forms for which D is negative, and indefinite forms for which D is positive. The case D = 0 or a positive square is rejected, because in that case the form breaks up into the product of two linear factors. It will be observed that ${\mbox{D}}\equiv 0,1\ ({\mbox{mod}}\ 4)$ according as b is even or odd; and that if $k^{2}$ is any odd square factor of D there will be forms of determinant D and divisor k.

If we write $x'=\alpha x+\beta y$, $y'=\gamma x+\delta y$, we have identically

$(a,b,c)(x',y')^{2}=(a',b',c')(x,y)^{2}$

where

$a'=a\alpha ^{2}+b\alpha \gamma +o\gamma ^{2}$

$b'=2a\alpha \beta +b(\alpha \delta +\beta \gamma )+2c\gamma \delta$

$c'=a\beta ^{2}+b\beta \delta +\delta ^{2}$

Hence also

${\mbox{D}}'=b'^{2}-4a'c'=(\alpha \delta -\beta \gamma )^{2}(b^{2}-4ac)=(\alpha \delta -\beta \gamma )^{2}{\mbox{D}}$.

Supposing that $\alpha ,\ \beta ,\ \gamma ,\ \delta$ are integers such that $\alpha \delta -\beta \gamma =n$, a number different from zero, $(a,\ b,\ c)$ is said to be transformed into $(a',\ b',\ c')$ by the substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$ of the nth order. If $n^{2}=1$, the two forms are said to be equivalent, and the equivalence is said to be proper or improper according as $n=1$ or $n=-1$. In the case of equivalence, not only are x', y' integers wherever x, y are so, but conversely; hence every number representable by (a, b, c) is representable by (a', b', c') and conversely. For the present we shall deal with proper equivalence only and write $f\sim f'$ to indicate that the forms f, f' are properly equivalent. Equivalent forms have the same divisor. A complete set of equivalent forms is said to form a class; classes of the same divisor are said to form an order, and of these the most important is the principal order, which consists of the primitive classes. It is a fundamental theorem that for a given determinant the number of classes is finite; this is proved by showing that every class must contain one at least of a certain finite number of so-called reduced forms, which can be found by definite rules of calculation.

33. Method of Reduction.—This differs according as D is positive or negative, and will require some preliminary lemmas. Suppose that any complex quantity $z=x+yi$ is represented in the usual way by a point (x, y) referred to rectangular axes. Then by plotting off all the points corresponding to $(\alpha z+\beta )/(\gamma z+\delta )$, we obtain a complete set of properly equivalent points. These all lie on the same side of the axis of x, and there is precisely one of them and no more which satisfies the conditions: (i.) that it is not outside the area which is bounded by the lines $2x=\pm 1$; (ii.) that it is not inside the circle $x^{2}+y^{2}=1$; (iii.) that it is not on the line $2x=1$, or on the arcs of the circle $x^{2}+y^{2}=1$ intercepted by $2x=1$ and $x=0$. This point will be called the reduced point equivalent to z. In the positive half-plane (y > 0) the aggregate of all reduced points occupies the interior and half the boundary of an area which will be called the fundamental triangle, because the areas equivalent to it, and finite, are all triangles bounded by circular arcs, and having angles 1/3π, 1/3π, 0 and the fundamental triangle may be considered as a special case when one vertex goes to infinity. The aggregate of equivalent triangles forms a kind of mosaic which fills up the whole of the positive half-plane. It will be convenient to denote the fundamental triangle (with its half-boundary, for which $x<0$) by ∇; for a reason which will appear later, the set of equivalent triangles will be said to make up the modular dissection of the positive half-plane.

Now let $f'=(a',b',c')$ be any definite form with a' positive and determinant — 𝚫. The root of $a'z^{2}+b'z+c'=0$ which is represented by a point in the positive half-plane is

$\omega ={-b'+i{\sqrt {\Delta }} \over 2a'}$

and this is a reduced point if either

${\begin{matrix}({\mbox{i.}})&b'<a'<c'\\({\mbox{ii.}})&b'=a',\ a'\leq c'\\({\mbox{iii.}})&a'=c',\ 0<b'\leq a'\end{matrix}}$

Cases (ii.) and (iii.) only occur when the representative point is on the boundary of ∇. A form whose representative point is reduced is said to be a reduced form. It follows from the geometrical theory that every form is equivalent to a reduced form, and that there are as many distinct classes of positive forms of determinant —∆ as there are reduced forms. The total number of reduced forms is limited, because in case (i.) we have $\Delta =4ac-b^{2}>3b^{2}$, so that $b<{\sqrt {\scriptstyle {1 \over 3}\Delta }}$, while $4a^{2}<4ac<\Delta +b^{2}<\scriptstyle {4 \over 3}\Delta$; in case (ii.) $\Delta =4ac-a^{2}>3a^{2}$, or else $a=b=c={\sqrt {\scriptstyle {1 \over 3}\Delta }}$; in case (iii.) $\Delta =4a^{2}-b^{2}>3b^{2},4a^{2}=\Delta +b^{2}<\scriptstyle {4 \over 3}\Delta$, or else $a=b=c={\sqrt {\scriptstyle {1 \over 3}\Delta }}$. With the help of these inequalities a complete set of reduced forms can be found by trial, and the number of classes determined. The latter cannot exceed 1/3∆; it is in general much less.

With an indefinite form (a, b, c) we may associate the representative circle

$a(x^{2}+y^{2})+bx+c=0,$

which cuts the axis of x in two real points. The form is said to be reduced if this circle cuts ∇; the condition for this is $a(a\pm \scriptstyle {1 \over 2}b+c)<0$, which can be expressed in the form $3a^{2}+(a\pm b)^{2}<{\mbox{D}}$, and it is hence clear that the absolute values of a, b, and therefore of c, are limited. As before, there are a limited number of reduced forms, but they are not all non-equivalent. In fact they arrange themselves, according to a law which is not very difficult to discover, in cycles or periods, each of which is associated with a particular class. The main result is the same as before: that the number of classes is finite, and that for each class we can find a representative form by a finite process of calculation.

34. Problem of Representation.—It is required to find out whether a given number m' can be represented by the given form $(a',\ b',\ c')$. One condition is clearly that the divisor of the form must be a factor of m'. Suppose this is the case; and let m, (a, b, c) be the quotients of m' and $(a',\ b',\ c')$ be the divisor in question. Then we have now to discover whether m can be represented by the primitive form (a, b, c). First of all we will consider proper representations

$m=(a,\ b,\ c)\ (\alpha ,\ \gamma )^{2}$

where 𝛼, 𝛾 are co-primes. Determine integers 𝛽, 𝛿 such that $\alpha \delta -\beta \gamma =1$, and apply to (a, b, c) the substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$; the new form will be (m, n, l), where

$n^{2}-4ml={\mbox{D}}=b^{2}-4ac$.

Consequently $n^{2}={\mbox{D}}\ ({\mbox{mod}}\ 4m)$, and D must be a quadratic residue of m. Unless this condition is satisfied, there is no proper representation of m by any form of determinant D. Suppose, however, that $n^{2}={\mbox{D}}\ ({\mbox{mod}}\ 4m)$ is soluble and that n₁, n₂, &c. are its roots. Taking any one of these, say n_i, we can find out whether (m, n_i, l_i) and (a, b, c) are equivalent; if they are, there is a substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$ which converts the latter into the former, and then $m=a\alpha ^{2}+b\alpha \gamma +c\gamma ^{2}$. As to derived representations, if $m=(a,\ b,\ c)\ (tx,\ ty)^{2}$, then m must have the square factor $t^{2}$, and $m/t^{2}=(a,\ b,\ c)\ (x,\ y)^{2}$; hence everything may be made to depend on proper representation by primitive forms.

35. Automorphs. The Pellian Equation.—A primitive form (a, b, c) is, by definition, equivalent to itself; but it may be so in more ways than one. In order that (a, b, c) may be transformed into itself by the substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$, it is necessary and sufficient that

${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}={\begin{pmatrix}{1 \over 2}(t+bu),&-cu\\au,&{1 \over 2}(t-bu)\end{pmatrix}}$

where (t, u) is an integral solution of

$t^{2}-{\mbox{D}}u^{2}=4$.

If D is negative and $-{\mbox{D}}>4$, the only solutions are $t=\pm 2,\ u=0$; ${\mbox{D}}=-3$ gives $(\pm 2,0),\ (\pm 1,\pm 1)$; ${\mbox{D}}=-4$ gives $(\pm 2,0),\ (0,\pm 1)$. On the other hand, if ${\mbox{D}}>0$ the number of solutions is infinite and if (t₁, u₁) is the solution for which t, u have their least positive ilalues, all the other positive solutions may be found from

${t_{n}+u_{n}{\sqrt {\mbox{D}}} \over 2}=\left({t_{1}+u_{1}{\sqrt {\mbox{D}}} \over 2}\right)^{n}\ (n=2,\ 3,\ 4\ .\ .\ .)$.

The substitutions by which (a, b, c) is transformed into itself are called its automorphs. In the case when ${\mbox{D}}=0\ ({\mbox{mod}}\ 4)$ we have $t=2{\mbox{T}}$, $u=2{\mbox{U}}$, ${\mbox{D}}=4{\mbox{N}}$, and (T, U) any solution of

${\mbox{T}}^{2}-{\mbox{NU}}^{2}=1$.

This is usually called the Pellian equation, though it should properly be associated with Fermat, who first perceived its importance. The minimum solution can be found by converting ${\sqrt {N}}$ into a periodic continued fraction.

The form (a, b, c) may be improperly equivalent to itself; in this case all its improper automorphs can be expressed in the form

${\begin{pmatrix}\lambda &,&(\kappa +b\lambda )/2a\\(\kappa -b\lambda )/2c&,&-\lambda \end{pmatrix}}$

where $\kappa ^{2}-{\mbox{D}}\lambda ^{2}=4ac$. In particular, if $b\equiv 0\ ({\mbox{mod}}\ a)$ the form (a, b, c) is improperly equivalent to itself. A form improperly equivalent to itself is said to be ambiguous.

36. Characters of a form or class. Genera.—Let $(a,\ b,\ c)$ be any primitive form; we have seen above (§ 32) that if $\alpha ,\ \beta ,\ \gamma ,\ \delta$ are any integers

$4(a\alpha ^{2}+b\alpha \gamma +c\gamma ^{2})(a\beta ^{2}+b\beta \delta +c\delta ^{2})=b'^{2}-(\alpha \delta -\beta \gamma )^{2}{\mbox{D}}$

where $b'=2a\alpha \beta +b(\alpha \delta +\beta \gamma )+2c\gamma \delta$. Now the expressions in brackets on the left hand may denote any two numbers m, n representable by the form $(a,\ b,\ c)$; the formula shows that 4mn is a residue of D, and hence mn is a residue of every odd prime factor of D, and if p is any such factor the symbols $\left({m \over p}\right)$ and $\left({n \over p}\right)$ will have the same value. Putting $(a,\ b,\ c)=f$, this common value is denoted by $\left({f \over p}\right)$ and called a quadratic character (or simply character) of f with respect to p. Since a is representable by $f(x=1,y=0)$ the value $\left({f \over p}\right)$ is the same as $\left({a \over p}\right)$. For example, if D = −140, the scheme of characters for the six reduced primitive forms, and therefore for the classes they represent, is

	$\left({f \over 5}\right)$	$\left({f \over 7}\right)$
(1, 0, 35) (4, ±2, 9)	+	+
(5, 0, 7) (3, ±2, 12)	−	−

In certain cases there are supplementary characters of the type $\left({-1 \over f}\right)$ and $\left({2 \over f}\right)$, and the characters $\left({f \over p}\right)$ are discriminated according as an odd or even power of p is contained in D; but in every case there are certain combinations of characters (in number one-half of all possible combinations) which form the total characters of actually existing classes. Classes which have the same total character are said to belong to the same genus. Each genus of the same order contains the same number of classes.

For any determinant D we have a principal primitive class for which all the characters are +; this is represented by the principal form (1, 0, −n) or (1, 1, −n) according as D is of the form 4n or 4n+1. The corresponding genus is called the principal genus. Thus, when D=−140, it appears from the table above that in the primitive order there are two genera, each containing three classes; and the non-existent total characters are $+-$; and $-+$.

37. Composition.—Considering X, Y as given lineo-linear functions of (x, y), (x', y') defined by the equations

${\mbox{X}}=p_{0}xx'+p_{1}xy'+p_{2}x'y+p_{3}yy'$

${\mbox{Y}}=q_{0}xx'+q_{1}xy'+q_{2}x'y+q_{3}yy'$

we may have identically, in x, y, x', y',

$(A,\ B,\ C)\ (X,\ Y)^{2}=(a,\ b,\ c)\ (x,\ y)^{2}\times (a',\ b',\ c')\ (x',\ y')^{2}$

and, this being so, the form (A, B, C) is said to be compounded of the two forms (a, b, c), (a', b', c'), the order of composition being indifferent. In order that two forms may admit of composition into a third, it is necessary and sufficient that their determinants be in the ratio of two squares. The most important case is that of two primitive forms 𝜙, 𝜒 of the same determinant; these can be compounded into a form denoted by 𝜙𝜒 or 𝜒𝜙 which is also primitive and of the same determinant as 𝜙 or 𝜒. If A, B, C are the classes to which 𝜙, 𝜒, 𝜙𝜒 respectively belong, then any form of A compounded with any form of B gives rise to a form belonging to C. For this reason we write C=AB=BA, and speak of the multiplication or composition of classes. The principal class is usually denoted by 1, because when compounded with any other class A it gives this same class A.

The total number of primitive classes being finite, h, say, the series A, A², A³, &c., must be recurring, and there will be a least exponent e such that ${\mbox{A}}^{e}=1$. This exponent is a factor of h, so that every class satisfies ${\mbox{A}}^{h}=1$. Composition is associative as well as commutative, that is to say, (AB)C=A(BC); hence the symbols A₁, A₂,. . . A_h for the h different classes define an Abelian group (see Groups) of order h, which is representable by one or more base-classes B₁, B₂, . . . B_i in such a way that each class A is enumerated once and only once by putting

$A=B_{1}^{x}B_{2}^{y}...B_{i}^{z}\qquad (x\leq m,\,y\leq n,\,.\,.\,.z\leq p)$

with $mn\,.\,.\,.\,p=h$, and $B_{1}^{m}=B_{2}^{n}=...=B_{i}^{p}=1$. Moreover, the bases may be so chosen that m is a multiple of n, n of the next corresponding index, and so on. The same thing may be said with regard to the symbols for the classes contained in the principal genus, because two forms of that genus compound into one of the same kind. If this latter group is cyclical, that is, if all the classes of the principal genus can be represented in the form 1, 𝖠, 𝖠²,. . .𝖠^𝑣−1, the determinant 𝖣 is said to be regular; if not, the determinant is irregular. It has been proved that certain specified classes of determinants are always irregular; but no complete criterion has been found, other than working out the whole set of primitive classes, and determining the group of the principal genus, for deciding whether a given determinant is irregular or not.

If 𝖠, 𝖡 are any two classes, the total character of 𝖠𝖡 is found by compounding the characters of 𝖠 and 𝖡. In particular, the class 𝖠², which is called the duplicate of 𝖠, always belongs to the principal genus. Gauss proved, conversely, that every class in the principal genus may be expressed as the duplicate of a class. An ambiguous class satisfies 𝖠²＝1, that is, its duplicate is the principal class; and the converse of this is true. Hence if 𝖡₁, 𝖡₂,. . .𝖡_𝑖 are the base-classes for the whole composition-group, and 𝖠＝𝖡₁^𝑥 𝖡₂^𝒚 . . . 𝖡_𝒊^𝒛 (as above) 𝖠＝1, if 2𝑥＝0 or 𝑚, 2𝑦＝0 or 𝑛, &c.; hence the number of ambiguous classes is 2^𝑖. As an example, when 𝖣＝−1460, there are four ambiguous classes, represented by

(1, 0, 365), (2, 2, 183), (5, 0, 73), (10, 10, 39);

hence the composition-group must be dibasic, and in fact, if we put 𝖡₁, 𝖡₂ for the classes represented by (11, 6, 34) and (2, 2, 183), we have 𝖡₁¹⁰＝𝖡₂²＝1 and the 20 primitive classes are given by 𝖡₁^𝒙B₂^𝒚(𝑥≤10, 𝑦≤2). In this case the determinant is regular and the classes in the principal genus are 1, 𝖡₁², 𝖡₁⁴, 𝖡₁⁶, 𝖡₁⁸.

38. On account of its historical interest, we may briefly consider the form 𝑥²+𝑦², for which 𝖣＝−4. If 𝑝 is an odd prime of the form 4𝑛+1, the congruence 𝑚²≡−4(mod 4𝑝) is soluble (§ 31); let one of its roots be 𝒎, and 𝑚²+4＝4𝑙𝑝. Then (𝑝, 𝑚, 𝑙) is of determinant −4, and, since there is only one primitive class for this determinant, we must have (𝑝, 𝑚, 𝑙)~(1, 0, 1). By known rules we can actually find a substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$ which converts the first form into the second; this being so, ${\begin{pmatrix}\delta ,&-\beta \\-\gamma ,&\alpha \end{pmatrix}}$ will transform the second into the first, and we shall have 𝑝＝𝛾²+𝛿², a representation of 𝑝 as the sum of two squares. This is unique, except that we may put 𝑝＝(±𝛾)²+(±𝛿)². We also have 2＝1²+1² while no prime 4𝑛+3 admits of such a representation.

The theory of composition for this determinant is expressed by the identity (𝑥²+𝑦²) (𝑥′²+𝑦′²)＝(𝑥𝑥′±𝒚𝒚′)²+(𝒙𝒚′∓𝒚𝒙′)²; and by repeated application of this, and the previous theorem, we can show that if 𝖭=2^𝒂𝒑^𝒃𝒒^𝒄. . ., where 𝒑, 𝒒,. . . are odd primes of the form 4𝒏+1, we can find solutions of 𝖭＝𝑥²+𝑦², and indeed distinct solutions. For instance 65＝1²+8²＝4²+7², and conversely two distinct representations 𝖭＝𝑥²+𝑦²＝𝑢²+𝑣² lead to the conclusion that 𝖭 is composite. This is a simple example of the application of the theory of forms to the difficult problem of deciding whether a given large number is prime or composite; an application first indicated by Gauss, though, in the present simple case, probably known to Fermat.

39. Number of classes. Class-number Relations.—It appears from Gauss’s posthumous papers that he solved the very difficult problem of finding a formula for ℎ(𝖣), the number of properly primitive classes for the determinant 𝖣. The first published solution, however, was that of P. G. L. Dirichlet; it depends on the consideration of series of the form ∑(𝑎𝑥²+𝑏𝑥𝑦+𝑐𝑦²)^−1−𝒔 where 𝒔 is a positive quantity, ultimately made very small. L. Kronecker has shown the connexion of Dirichlet’s results with the theory of elliptic functions, and obtained more comprehensive formulae by taking (𝒂, 𝒃, 𝒄) as the standard type of a quadratic form, whereas Gauss, Dirichlet, and most of their successors, took (𝒂, 2𝒃, 𝒄) as the standard, calling (𝒃²-𝒂𝒄) its determinant. As a sample of the kind of formulae that are obtained, let 𝑝 be a prime of the form 4𝑛+3; then

$h(-4p)=\Sigma \alpha -\Sigma \beta$, $\qquad h(4p)\log(t+u{\sqrt {p}})=\log \Pi \left(\tan {b\pi \over 4p}\right)$

where in the first formula 𝝨𝛼 means the sum of all quadratic residues of 𝑝 contained in the series 1, 2, 3,. . .1/2(𝑝∼1) and 𝝨𝛽 is the sum of the remaining non-residues; while in the second formula (𝒕, 𝒖) is the least positive solution of 𝒕²−𝒑𝒖²＝1, and the product extends to all values of 𝑏 in the set 1, 3, 5,. . .(4𝑝−1) of which 𝑝 is a non-residue. The remarkable fact will be noticed that the second formula gives a solution of the Pellian equation in a trigonometrical form.

Kronecker was the first to discover, in connexion with the complex multiplication of elliptic functions, the simplest instances of a very curious group of arithmetical formulae involving sums of class-numbers and other arithmetical functions; the theory of these relations has been greatly extended by A. Hurwitz. The simplest of all these theorems may be stated as follows. Let 𝖧 (𝚫) represent the number of classes for the determinant −𝚫, with the convention that 1/2 and not 1 is to be reckoned for each class containing a reduced form of the type (𝑎, o, 𝑎) and 1/3 for each class containing a reduced form (𝑎, 𝑎, 𝑎); then if 𝑛 is any positive integer,

$\sum _{\kappa =0,\pm 1,...}H(4n-\kappa ^{2})=\Phi (n)+\Psi (n)\qquad \qquad (-2{\sqrt {n}}\leq \kappa \leq 2{\sqrt {n}})$

where 𝚽(𝑛) means the sum of the divisors of 𝑛, and 𝚿(𝑛) means the excess of the sum of those divisors of 𝑛 which are greater than ${\sqrt {n}}$ over the sum of those divisors which are less than ${\sqrt {n}}$. The formula is obtained by calculating in two different ways the number of reduced values of 𝑧 which satisfy the modular equation 𝖩(𝒏𝒛)＝𝖩(𝒛),

where 𝖩(𝒛) is the absolute invariant which, for the elliptic function 𝔭(𝑢; 𝑔₂, 𝑔₃) is 𝑔₂³÷(𝑔₂³−27𝑔₃²), and 𝑧 is the ratio of any two primitive periods taken so that the real part of 𝑖𝑧 is negative (see below, § 68). It should be added that there is a series of scattered papers by J. Liouville, which implicitly contain Kronecker’s class-number relations, obtained by a purely arithmetical process without any use of transcendents.

40. Bilinear Forms.—A bilinear form means an expression of the type 𝚺𝒂_𝒊𝒌𝒙_𝒊𝒚_𝒌 (𝒊＝1, 2,. . .𝒎; 𝒌＝1, 2,. . .𝒏); the most important case is when 𝒎＝𝒏, and only this will be considered here. The invariants of a form are its determinant [𝑎_𝑛𝑛] and the elementary factors thereof. Two bilinear forms are equivalent when each can be transformed into the other by linear integral substitutions 𝑥′＝𝚺𝛼𝑥, 𝒚′＝𝚺𝛽𝒚. Every bilinear form is equivalent to a reduced form $\sum _{1}^{r}e_{i}x_{i}y_{i}$, and 𝒓＝𝒏, unless [𝑎_𝑛𝑛]＝0. In order that two forms may be equivalent it is necessary and sufficient that their invariants should be the same. Moreover, if 𝑎∼𝑏 and 𝑐∼𝑑, and if the invariants of the forms 𝑎+𝜆𝑐, 𝑏+𝜆𝑑 are the same for all values of 𝜆, we shall have 𝑎+𝜆𝑐∼𝑏+𝜆𝑑, and the transformation of one form to the other may be effected by a substitution which does not involve 𝜆. The theory of bilinear forms practically includes that of quadratic forms, if we suppose 𝒙_𝒊, 𝒚_𝒊 to be cogredient variables. Kronecker has developed the case when 𝒏＝2, and deduced various class-relations for quadratic forms in a manner resembling that of Liouville. So far as the bilinear forms are concerned, the main result is that the number of classes for the positive determinant 𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁＝𝚫 is 12{𝚽(𝚫)+𝚿(𝚫)}+2𝜖, where 𝜖 is 1 or 0 according as 𝚫 is or is not a square, and the symbols 𝚽, 𝚿 have the meaning previously assigned to them (§ 39).

41. Higher Quadratic Forms.—The algebraic theory of quadratics is so complete that considerable advance has been made in the much more complicated arithmetical theory. Among the most important results relating to the general case of n variables are the proof that the class-number is finite; the enumeration of the arithmetical invariants of a form; classification according to orders and genera, and proof that genera with specified characters exist; also the determination of all the rational transformations of a given form into itself. In connexion with a definite form there is the important conception of its weight; this is defined as the reciprocal of the number of its proper automorphs. Equivalent forms are of the same weight; this is defined to be the weight of their class. The weight of a genus or order is the sum of the weights of the classes contained in it; and expressions for the weight of a given genus have actually been obtained. For binary forms the sum of the weights of all the genera coincides with the expression denoted by H(𝚫) in § 39. The complete discussion of a form requires the consideration of (𝒏−2) associated quadratics; one of these is the contravariant of the given form, each of the others contains more than 𝒏 variables. For certain quaternary and senary classes there are formulae analogous to the class-relations for binary forms referred to in § 39 (see Smith, Proc. R.S. xvi., or Collected Papers, i. 510).

Among the most interesting special applications of the theory are certain propositions relating to the representation of numbers as the sum of squares. In order that a number may be expressible as the sum of two squares it is necessary and sufficient for it to be of the form 𝖯𝖰², where 𝖯 has no square factor and no prime factor of the form 4𝑛+3. A number is expressible as the sum of three squares if, and only if, it is of the form 𝑚²𝑛 with 𝑛≡1, ±2, ±3 (mod 8); when 𝑚＝1 and 𝑛≡3 (mod 8), all the squares are odd, and hence follows Fermat’s theorem that every number can be expressed as the sum of three triangular numbers (one or two of which may be 0). Another famous theorem of Fermat’s is that every number can be expressed as the sum of four squares; this was first proved by Jacobi, who also proved that the number of solutions of 𝑛＝𝑥²+𝑦²+𝑧²+𝑡² is 8𝚽(𝒏), if 𝒏 is odd, while if 𝒏 is even it is 24 times the sum of the odd factors of 𝒏. Explicit and finite, though more complicated, formulae have been obtained for the number of representations of 𝒏 as the sum of five, six, seven and eight squares respectively. As an example of the outstanding difficulties of this part of the subject may be mentioned the problem of finding all the integral (not merely rational) automorphs of a given form 𝒇. When 𝒇 is ternary, C. Hermite has shown that the solution depends on finding all the integral solutions of 𝖥(𝒙, 𝒚, 𝒛)+𝒕²＝1, where 𝖥 is the contra variant of 𝒇.

Thanks to the researches of Gauss, Eisenstein, Smith, Hermite and others, the theory of ternary quadratics is much less incomplete than that of quadratics with four or more variables. Thus methods of reduction have been found both for definite and for indefinite forms; so that it would be possible to draw up a table of representative forms, if the result were worth the labour. One specially important theorem is the solution of 𝒂𝒙²+𝒃𝑦²+𝒄𝒛²＝0; this is always possible if −𝑏𝑐, −𝑐𝑎, −𝑎𝑏 are quadratic residues of 𝑎, 𝑏, 𝑐 respectively, and a formula can then be obtained which furnishes all the solutions.

42. Complex Numbers.—One of Gauss’s most important and far-reaching contributions to arithmetic was his introduction of complex integers 𝑎＋𝑏𝑖, where 𝑎, 𝑏 are ordinary integers, and, as usual, 𝒊²＝－1. In this theory there are four units ±1, ±𝒊; the numbers 𝒊^𝒉(𝒂+𝒃𝒊) are said to be associated; 𝒂－𝒃𝒊 is the conjugate of 𝑎＋𝑏𝑖 and we write 𝖭(𝒂±𝒃)＝𝑎²＋𝑏², the norm of 𝑎＋𝑏𝑖, its conjugate, and associates. The most fundamental proposition in the theory is that the process of residuation (§ 24) is applicable; namely, if 𝑚, 𝑛 are any two complex integers and 𝖭(𝑚)＞𝖭(𝑛), we can always find integers 𝑞, 𝑟 such that 𝑚＝𝑞𝑛＋𝑟 with 𝖭(𝑟)⩽1/2𝖭(𝑛). This may be proved analytically, but is obvious if we mark complex integers by points in a plane. Hence immediately follow propositions about resolutions into prime factors, greatest common measure, &c., analogous to those in the ordinary theory; it will only be necessary to indicate special points of difference.

We have 2 ＝ －𝑖(1+𝑖)², so that 2 is associated with a square; a real prime of the form 4𝒏＋3 is still a prime but one of the form 4𝒏＋1 breaks up into two conjugate prime factors, for example 5 ＝ (1－2𝑖)(1＋2𝑖). An integer is even, semi-even, or odd according as it is divisible by (1＋𝑖)², (1＋𝑖) or is prime to (1＋𝑖). Among four associated odd integers there is one and only one which≡1 (mod 2＋2𝒊); this is said to be primary; the conjugate of a primary number is primary, and the product of any number of primaries is primary. The conditions that 𝒂＋𝒃𝒊 may be primary are b≡0 (mod 2), 𝑎＋𝑏−1≡0 (mod 4). Every complex integer can be uniquely expressed in the form 𝑖^𝑚(1＋𝑖)^𝑛𝑎^𝛼𝑏^𝛽𝑐^𝛾 . . ., where 0⩽𝑚＜4, and 𝒂, 𝒃, 𝒄, . . . are primary primes.

With respect to a complex modulus 𝑚, all complex integers may be distributed into 𝖭 (𝑚) incongruous classes. If 𝑚＝ℎ(𝒂＋𝒃𝑖) where 𝒂, 𝒃 are co-primes, we may take as representatives of these classes the residues 𝑥＋𝑦𝑖 where 𝑥＝0, 1, 2, . . . {(𝒂²＋𝒃²)ℎ－1}; 𝑦＝0, 1, 2, . . . (ℎ－1). Thus when 𝒃＝0 we may take 𝑥＝0, 1, 2, . . . (ℎ－1); 𝑦＝0, 1, 2, . . . (ℎ－1), giving the ℎ² residues of the real number ℎ; while if 𝒂＋𝒃𝑖 is prime, 1, 2, 3, . . . (𝒂²＋𝒃²－1) form a complete set of residues.

The number of residues of 𝑚 that are prime to 𝑚 is given by

$\phi (m)={\mbox{N}}(m)\Pi \left(1-{1 \over {\mbox{N}}(p)}\right)$

where the product extends to all prime factors of 𝑚. As an analogue to Fermat’s theorem we have, for any integer prime to the modulus,

𝑥^𝜙(𝑚)≡1(mod 𝑚), 𝑥^{𝖭(𝒑)－1}≡1 (mod 𝒑)

according as 𝑚 is composite or prime. There are 𝜙{𝖭(𝒑)－1)} primitive roots of the prime 𝒑; a primitive root in the real theory for a real prime 4𝒏＋1 is also a primitive root in the new theory for each prime factor of (4𝒏＋1), but if 𝒑＝4𝒏＋3 be a prime its primitive roots are necessarily complex.

43. If 𝒑, 𝒒 are any two odd primes, we shall define the symbols $\left({\frac {p}{q}}\right)_{2}$ and $\left({\frac {p}{q}}\right)_{4}$ by the congruences

$p^{\scriptstyle {1 \over 2}\{N(q)-1\}}\equiv \left({p \over q}\right)_{2}$, $\ \ p^{\scriptstyle {1 \over 4}\{N(q)-1\}}\equiv \left({p \over q}\right)_{4}$

it being understood that the symbols stand for absolutely least residues. It follows that $\left({p \over q}\right)_{2}=1$ or −1 according as 𝒑 is a quadratic residue of 𝒒 or not; and that $\left({p \over q}\right)_{4}=1$ only if 𝒑 is a biquadratic residue of 𝒒. If 𝒑, 𝒒 are primary primes, we have two laws of reciprocity, expressed by the equations

$\left({p \over q}\right)_{2}=\left({q \over p}\right)_{2}$, $\ \ \left({p \over q}\right)_{4}\left({q \over p}\right)_{4}=(-1)^{{1 \over 4}\{N(p)-1\}.{1 \over 4}\{.\ N(q)-1\}}$

To these must be added the supplementary formulae

$\qquad \qquad \qquad \qquad \left({i \over p}\right)_{2}=(-1)^{{1 \over ?}\{N(p)-1\}},$	$\ \ \left({1+i \over a+bi}\right)_{2}=(-1)^{{1 \over 8}\{(a+b)^{2}-1\}}$
$\left({i \over a+bi}\right)_{4}=i^{{1 \over 2}(a-1)}$	$\left({1+i \over a+bi}\right)_{4}=i^{{1 \over 4}\{a+b-(1+b)^{2}\}}$

𝒂＋𝒃𝑖 being a primary odd prime. In words, the law of biquadratic reciprocity for two primary odd primes may be expressed by saying that the biquadratic characters of each prime with respect to the other are identical, unless 𝒑 ＝ 𝒒 ≡ 3 ＋ 2𝑖 (mod 4), in which case they are opposite. The law of biquadratic reciprocity was discovered by Gauss, who does not seem, however, to have obtained a complete proof of it. The first published proof is that of Eisenstein, which is very beautiful and simple, but involves the theory of lemniscate functions. A proof on the lines indicated in Gauss’s posthumous papers has been developed by Busche; this probably admits of simplification. Other demonstrations, for instance Jacobi’s, depend on cyclotomy (see below).

44. Algebraic Numbers.—The first extension of Gauss’s complex theory was made by E. E. Kummer, who considered complex numbers represented by rational integral functions of any roots of unity, thus including the ordinary theory and Gauss’s as special cases. He was soon faced by the difficulty that, in some cases, the law that an integer can be uniquely expressed as the product of prime factors appeared to break down. To see how this happens take the equation 𝜂²＋𝜂＋6＝0, the roots of which are expressible as rational integral functions of 23rd roots of unity, and let 𝜂 be either of the roots. If we define 𝒂𝜂＋𝒃 to be an integer, when 𝒂, 𝒃 are natural numbers, the product of any number of such integers is uniquely expressible in the form 𝑙𝜂＋𝑚. Conversely every integer can be expressed as the product of a finite number of indecomposable integers 𝒂＋𝒃𝜂, that is, integers which cannot be further resolved into factors of the same type. But this resolution is not necessarily unique: for instance 6＝2.3＝－𝜂(𝜂＋1), where 2, 3, 𝜂, 𝜂＋1 are all indecomposable and essentially distinct. To see the way in which Kummer surmounted the difficulty consider the congruence

𝑢²＋𝑢＋6≡0(mod 𝒑)

where 𝒑 is any prime, except 23. If －23𝖱𝒑 this has two distinct roots 𝑢₁, 𝑢₂; and we say that 𝒂𝜂＋𝒃 is divisible by the ideal prime factor of 𝒑 corresponding to 𝑢₁, if 𝒂𝑢₁＋𝒃≡0 (mod 𝒑). For instance, if 𝒑＝2 we may put 𝑢₁＝0, 𝑢₂＝1 and there will be two ideal factors of 2, say 𝒑₁ and 𝒑₂ such that 𝒂𝜂＋𝒃≡0 (mod 𝒑₁) if 𝒃≡0 (mod 2) and 𝒂𝜂＋𝒃≡0 (mod 𝒑₂) if 𝒂＋𝒃≡0 (mod 2). If both these congruences are satisfied, 𝒂≡𝒃≡0 (mod 2) and 𝒂𝜂＋𝒃 is divisible by 2 in the ordinary sense. Moreover (𝒂𝜂＋𝒃)(𝒄𝜂＋𝑑)＝(𝒃𝒄＋𝒂𝑑－𝒂𝒄)𝜂＋(𝒃𝑑－6𝒂𝒄) and if this product is divisible by 𝒑₁, 𝒃𝑑≡0 (mod 2), whence either 𝒂𝜂＋𝒃 or 𝒄𝜂＋𝑑 is divisible by 𝒑₁; while if the product is divisible by 𝒑₂ we have 𝒃𝒄＋𝒂𝑑＋𝒃𝑑－7𝒂𝒄＝0 (mod 2) which is equivalent to (𝒂＋𝒃)(𝒄＋𝑑)≡0 (mod 2), so that again either 𝒂𝜂＋𝒃 or 𝒄𝜂＋𝑑 is divisible by 𝒑₂. Hence we may properly speak of 𝒑₁ and 𝒑₂ as prime divisors. Similarly the congruence 𝑢²＋𝑢＋6≡0 (mod 3) defines two ideal prime factors of 3, and 𝒂𝜂＋𝒃 is divisible by one or the other of these according as 𝒃≡0 (mod 3) or 2𝒂＋𝒃≡0 (mod 3); we will call these prime factors 𝒑₃, 𝒑₄. With this notation we have (neglecting unit factors)

2＝𝒑₁𝒑₂, 3＝𝒑₃𝒑₄, 𝜂＝𝒑₁𝒑₃, 1＋𝜂＝𝒑₂𝒑₄

Real primes of which －23 is a non-quadratic residue are also primes in the field (𝜂); and the prime factors of any number 𝒂𝜂＋𝒃, as well as the degree of their multiplicity, may be found by factorizing (6𝒂²－𝒂𝒃＋𝒃²), the norm of (𝒂𝜂＋𝒃). Finally every integer divisible by 𝒑₂ is expressible in the form ±2𝑚±(1+𝜂)𝒏 where 𝑚, 𝒏 are natural numbers (or zero); it is convenient to denote this fact by writing 𝒑₂＝[2, 1＋𝜂], and calling the aggregate 2𝑚＋(1＋𝜂)𝒏 a compound modulus with the base 2, 1＋𝜂. This generalized idea of a modulus is very important and far-reaching; an aggregate is a modulus when, if 𝛼, 𝛽 are any two of its elements, 𝛼＋𝛽 and 𝛼－𝛽 also belong to it. For arithmetical purposes those moduli are most useful which can be put into the form [𝛼₁ , 𝛼₂,…𝛼_𝒏] which means the aggregate of all the quantities 𝑥₁𝛼₁＋𝑥₂𝛼₂＋…＋𝑥_𝒏𝛼_𝒏 obtained by assigning to (𝑥₁,𝑥₂,…𝑥_𝒏), independently, the values 0₁±1, ±2, &c. Compound moduli may be multiplied together, or raised to powers, by rules which will be plain from the following example. We have
𝒑₂²＝[4, 2(1+𝜂), (1+𝜂)²]＝[4, 2＋2𝜂,－5＋𝜂]＝[4, 12,－5＋𝜂]
    ＝[4,－5＋𝜂]＝[4, 3＋𝜂]
hence
𝒑₂³＝𝒑₂².𝒑₂＝[4, 3＋𝜂]×[2, 1＋𝜂]＝[8, 4＋4𝜂, 6＋2𝜂, 3＋4𝜂＋𝜂²]
    ＝[8, 4＋4𝜂, 6＋2𝜂, －3＋3𝜂]＝(𝜂－1)[𝜂＋2, 𝜂－6, 3]＝(𝜂－1)[1, 𝜂]
Hence every integer divisible by 𝒑₂³ is divisible by the actual integer (𝜂－1) and conversely; so that in a certain sense we may regard 𝒑₂ as a cube root. Similarly the cube of any other ideal prime is of the form (𝒂𝜂+𝒃)[1, 𝜂]. According to a principle which will be explained further on, all primes here considered may be arranged in three classes; one is that of the real primes, the others each contain ideal primes only. As we shall see presently all these results are intimately connected with the fact that for the determinant －23 there are three primitive classes, represented by (1, 1, 6) (2, 1, 3), (2, －1, 3) respectively.

45. Kummer’s definition of ideal primes sufficed for his particular purpose, and completely restored the validity of the fundamental theorems about factors and divisibility. His complex integers were more general than any previously considered and suggested a definition of an algebraic integer in general, which is as follows: if 𝒂₁,𝒂₂,…𝒂_𝒏 are ordinary integers (i.e. elements of N, § 7), and 𝜃 satisfies an equation of the form

𝜃＋𝒂₁𝜃^𝒏－1＋𝒂₂𝜃^𝒏－2＋ . . . +𝒂_𝒏－1𝜃＋𝒂_𝒏＝0

𝜃 is said to be an algebraic integer. We may suppose this equation irreducible; 𝜃 is then said to be of the 𝒏th order. The 𝒏 roots 𝜃, 𝜃′, 𝜃″,. . .𝜃^(𝒏－1) are all different, and are said to be conjugate. If the equation began with 𝒂₀𝜃^𝒏 instead of 𝜃^𝒏, 𝜃 would still be an algebraic number; every algebraic number can be put into the form 𝜃 ∕𝑚, where 𝑚 is a natural number and 𝜃 an algebraic integer.

Associated with 𝜃 we have a field (or corpus) Ω＝𝖱(𝜃) consisting of all rational functions of 𝜃 with real rational coefficients; and in like manner we have the conjugate fields Ω′＝𝖱(𝜃′), &c. The aggregate of integers contained in Ω is denoted by ο.

Every element of Ω can be put into the form

𝜔＝𝒄₀＋𝒄₁𝜃＋ . . . ＋𝒄_{𝒏 —1}𝜃^𝒏—1

where 𝒄₀, 𝒄₁,…𝒄_𝒏—1 are real and rational. If these coefficients are all integral, 𝜔 is an integer; but the converse is not necessarily true. It is possible, however, to find a set of integers 𝜔₁, 𝜔₂,…𝜔_𝒏, belonging to Ω, such that every integer in Ω can be uniquely expressed in the form

𝜔＝ℎ₁𝜔₁＋ℎ₂𝜔₂＋ . . . ＋ ℎ_𝒏𝜔_𝒏

where $h_{1},h_{2},\dots h_{n}$ are elements of $\mathrm {N}$ which may be called the

co-ordinates of $\omega$ with respect to the base $\omega _{1},\omega _{2},\dots \omega _{n}$. Thus ${\mathfrak {o}}$ is a modulus (§ 44), and we may write ${\mathfrak {o}}=[\omega _{1},\omega _{2},\dots \omega _{n}]$. Having found one base, we can construct any number of equivalent bases by means of equations such as ${\omega _{i}}'={\underset {i}{\Sigma }}c_{ij}\omega _{j}$, where the rational integral coefficients $c_{ij}$ are such that the determinant $|c_{nn}|=\pm 1$.

If we write

$\surd \Delta ={\begin{vmatrix}\omega _{1},&\omega _{2},&\dots &\omega _{n}\\{\omega '}_{1},&{\omega '}_{2},&\dots &{\omega '}_{n}\\{\omega ''}_{1},&{\omega ''}_{2},&\dots &{\omega ''}_{n}\\\vdots \\{\omega ^{(n-1)}}_{1},&{\omega ^{(n-1)}}_{2},&\dots &{\omega ^{(n-1)}}_{n}\\\end{vmatrix}}$

$\Delta$ is a rational integer called the discriminant of the field. Its value is the same whatever base is chosen.

If $\alpha$ is any integer in $\Omega$, the product of $\alpha$ and its conjugates is a rational integer called the norm of $\alpha$, and written $\mathrm {N} (\alpha )$. By considering the equation satisfied by $\alpha$ we see that $\mathrm {N} (\alpha )=\alpha \alpha _{1}$ where $\alpha _{1}$ is an integer in $\Omega$. It follows from the definition that if $\alpha ,\beta$ are any two integers in $\Omega$, then $\mathrm {N} (\alpha \beta )=\mathrm {N} (\alpha )\mathrm {N} (\beta )$; and that for an ordinary real integer $m$, we have $\mathrm {N} (m)=m^{n}$.

46. Ideals.—The extension of Kummer’s results to algebraic numbers in general was independently made by J. W. R. Dedekind and Kronecker; their methods differ mainly in matters of notation and machinery, each having special advantages of its own for particular purposes. Dedekind’s method is based upon the notion of an ideal, which is defined by the following properties:—

(i.) An ideal ${\mathfrak {m}}$ is an aggregate of integers in $\Omega$.

(ii.) This aggregate is a modulus; that is to say, if $\mu ,\mu '$ are any two elements of ${\mathfrak {m}}$ (the same or different) $\mu -\mu '$ is contained in ${\mathfrak {m}}$. Hence also ${\mathfrak {m}}$ contains a zero element, and $\mu +\mu '$ is an element of ${\mathfrak {m}}$.

(iii.) If $\mu$ is any element of ${\mathfrak {m}}$, and $\omega$ any element of ${\mathfrak {o}}$, then $\omega \mu$ is an element of ${\mathfrak {m}}$. It is this property that makes the notion of an ideal more specific than that of a modulus.

It is clear that ideals exist; for instance, ${\mathfrak {o}}$ itself is an ideal. Again, all integers in $\Omega$ which are divisible by a given integer $\alpha$ (in ${\mathfrak {o}}$) form an ideal; this is called a principal ideal, and is denoted by ${\mathfrak {o}}\alpha$. Every ideal can be represented by a base (§§ 44, 45), so that we may write ${\mathfrak {m}}=[\mu _{1},\mu _{2},\dots \mu _{n}]$, meaning that every element of ${\mathfrak {m}}$ can be uniquely expressed in the form $\Sigma h_{i}\mu _{i}$, where $h_{i}$ is a rational integer. In other words, every ideal has a base (and therefore, of course, an infinite number of bases). If ${\mathfrak {a}},{\mathfrak {b}}$ are any two ideals, and if we form the aggregate of all products $\alpha \beta$ obtained by multiplying each element of the first ideal by each element of the second, then this aggregate, together with all sums of such products, is an ideal which is called the product of ${\mathfrak {a}}$ and ${\mathfrak {b}}$ and written ${\mathfrak {ab}}$ or ${\mathfrak {ba}}$. In particular ${\mathfrak {oa}}={\mathfrak {a}},{\mathfrak {o}}^{2}={\mathfrak {o}},{\mathfrak {o}}\alpha \ .\ {\mathfrak {o}}\beta ={\mathfrak {o}}\alpha \beta$. This law of multiplication is associative as well as commutative. It is clear that every element of ${\mathfrak {ab}}$ is contained in ${\mathfrak {a}}$: it can be proved that, conversely, if every element of ${\mathfrak {c}}$ is contained in ${\mathfrak {a}}$, there exists an ideal ${\mathfrak {b}}$ such that ${\mathfrak {ab}}={\mathfrak {c}}$. In particular, if $\alpha$ is any element of ${\mathfrak {a}}$, there is an ideal ${\mathfrak {a}}'$ such that ${\mathfrak {o}}\alpha ={\mathfrak {aa}}'$. A prime ideal is one which has no divisors except itself and ${\mathfrak {o}}$. It is a fundamental theorem that every ideal can be resolved into the product of a finite number of prime ideals, and that this resolution is unique. It is the decomposition of a principal ideal into the product of prime ideals that takes the place of the resolution of an integer into its prime factors in the ordinary theory. It may happen that all the ideals in $\Omega$ are principal ideals; in this case every resolution of an ideal into factors corresponds to the resolution of an integer into actual integral factors, and the introduction of ideals is unnecessary. But in every other case the introduction of ideals or some equivalent notion, is indispensable. When two ideals have been resolved into their prime factors, their greatest common measure and least common multiple are determined by the ordinary rules. Every ideal may be expressed (in an infinite number of ways) as the greatest common measure of two principal ideals.

47. There is a theory of congruences with respect to an ideal modulus. Thus $\alpha \equiv \beta {\pmod {\mathfrak {m}}}$ means that $\alpha -\beta$ is an element of ${\mathfrak {m}}$. With respect to ${\mathfrak {m}}$, all the integers in $\Omega$ may be arranged in a finite number of incongruent classes. The number of these classes is called the norm of ${\mathfrak {m}}$, and written $\mathrm {N} ({\mathfrak {m}})$. The norm of a prime ideal ${\mathfrak {p}}$ is some power of a real prime $p$; if $\mathrm {N} ({\mathfrak {p}})=p^{f}$, ${\mathfrak {p}}$ is said to be a prime ideal of degree $f$. If ${\mathfrak {m}},{\mathfrak {n}}$ are any two ideals, then $\mathrm {N} (mn)=\mathrm {N} ({\mathfrak {m}})\mathrm {N} ({\mathfrak {n}})$. If $\mathrm {N} ({\mathfrak {m}})=m$, then $m\equiv 0{\pmod {\mathfrak {m}}}$, and there is an ideal ${\mathfrak {m'}}$ such that ${\mathfrak {o}}m={\mathfrak {mm}}'$. The norm of a principal ideal ${\mathfrak {o}}\alpha$ is equal to the absolute value of $\mathrm {N} (\alpha )$ as defined in § 45.

The number of incongruent residues prime to ${\mathfrak {m}}$ is—

$\phi ({\mathfrak {m}})=\mathrm {N} ({\mathfrak {m}})\Pi \left(1-{\frac {1}{\mathrm {N} ({\mathfrak {p}})}}\right)$,

where the product extends to all prime factors of ${\mathfrak {m}}$. If $\omega$ is any element of ${\mathfrak {o}}$ prime to ${\mathfrak {m}}$,

$\omega ^{\phi ({\mathfrak {m}})}\equiv 1{\pmod {\mathfrak {m}}}$.

Associated with a prime modulus ${\mathfrak {p}}$ for which $\mathrm {N} ({\mathfrak {p}})=p^{f}$ we have $\phi (p^{f}-1)$ primitive roots, where $\phi$ has the meaning given to it in the ordinary theory. Hence follow the usual results about exponents, indices, solutions of linear congruences, and so on. For any modulus ${\mathfrak {m}}$ we have $\mathrm {N} ({\mathfrak {m}})=\Sigma \phi ({\mathfrak {d}})$, where the sum extends to all the divisors of ${\mathfrak {m}}$.

48. Every element of ${\mathfrak {o}}$ which is not contained in any other ideal is an algebraic unit. If the conjugate fields $\Omega ,\Omega ',\dots \Omega ^{}(n-1)$ consist of $r_{1}$ real and $2r_{2}$ imaginary fields, there is a system of units $\epsilon _{1},\epsilon _{2},\dots \epsilon _{r}$, where $r=r_{1}+r_{2}-1$, such that every unit in $\Omega$ is expressible in the form $\epsilon =\rho {\epsilon _{1}}^{a}{\epsilon _{2}}^{b}\dots {\epsilon _{r}}^{l}$ where $\rho$ is a root of unity contained in $\Omega$ and $a,b,\dots l$ are natural numbers. This theorem is due to Dirichlet.

The norm of a unit is $+1$ or $-1$; and the determination of all the units contained in a given field is in fact the same as the solution of a Diophantine equation

$\mathrm {F} (h_{1},h_{2},\dots h_{n})=\pm 1$.

For a quadratic field the equation is of the form ${h_{1}}^{2}-n{h_{2}}^{2}=\pm 1$, and the theory of this is complete; but except for certain special cubic corpora little has been done towards solving the important problem of assigning a definite process by which, for a given field, a system of fundamental units may be calculated. The researches of Jacobi, Hermite, and Minkowsky seem to show that a proper extension of the method of continued fractions is necessary.

49. Ideal Classes.—If ${\mathfrak {m}}$ is any ideal, another ideal ${\mathfrak {n}}$ can always be found such that ${\mathfrak {mn}}$ is a principal ideal; for instance, one such multiplier is ${\mathfrak {m}}^{-1}\mathrm {N} ({\mathfrak {m}})$. Two ideals ${\mathfrak {m}},{\mathfrak {m}}'$ are said to be equivalent (${\mathfrak {m}}\sim {\mathfrak {m}}'$) or to belong to the same class, if there is an ideal ${\mathfrak {n}}$ such that ${\mathfrak {mn}},{\mathfrak {m'n}}$ are both principal ideals. It can be proved that two ideals each equivalent to a third are equivalent to each other and that all ideals in $\Omega$ may be distributed into a finite number, $h$, of ideal classes. The class which contains all principal ideals is called the principal class and denoted by $\mathrm {O}$.

If ${\mathfrak {m}},{\mathfrak {n}}$ are any two ideals belonging to the classes $\mathrm {A} ,\mathrm {B}$ respectively, then ${\mathfrak {mn}}$ belongs to a definite class which depends only upon $\mathrm {A} ,\mathrm {B}$ and may be denoted by $\mathrm {AB}$ or $\mathrm {BA}$ indifferently. Thus the class-symbols form an Abelian group of order $h$, of which $\mathrm {O}$ is the unit element; and, mutatis mutandis, the theorems of § 37 about composition of classes still hold good.

The principal theorem with regard to the determination of $h$ is the following, which is Dedekind’s generalization of the corresponding one for quadratic fields, first obtained by Dirichlet. Let

$\zeta (s)={\underset {({\mathfrak {m}})}{\Sigma }}\mathrm {N} ({\mathfrak {m}})-^{s}$

where the sum extends to all ideals ${\mathfrak {m}}$ contained in $\Omega$; this converges so long as the real quantity $s$ is positive and greater than $1$. Then $\kappa$ being a certain quantity which can be calculated when a fundamental system of units is known, we shall have

$\kappa h={\underset {s=1}{Lt}}\{(s-1)\zeta (s)\}$.

The expression for $\kappa$ is rather complicated, and very peculiar; it may be written in the form

$\kappa ={\frac {2^{r_{1}+r_{2}}\pi ^{r_{2}}}{w}}.{\frac {\mathrm {R} }{|\surd \Delta |}}$

where $|\surd \Delta |$ means the absolute value of the square root of the discriminant of the field, $r_{1},r_{2}$ have the same meaning as in § 48, $w$ is the number of roots of unity in $\Omega$, and $\mathrm {R}$ is a determinant of the form $|l_{i}(\epsilon ^{j})|$, of order $(r_{1}+r_{2}-1)$, with elements which are, in a certain special sense, “logarithms” of the fundamental units $\epsilon _{1},\epsilon _{2},\dots \epsilon _{r}$.

50. The discriminant $\Delta$ enjoys some very remarkable properties. Its value is always different from $\pm 1$; there can be only a finite number of fields which have a given discriminant; and the rational prime factors of $\Delta (\Omega )$ are precisely those rational primes which, in $\Omega$, are divisible by the square (or some higher power) of a prime ideal. Consequently, every rational prime not contained in $\Delta$ is resolvable, in $\Omega$, into the product of distinct primes, each of which occurs only once. The presence of multiple prime factors in the discriminant was the principal difficulty in the way of extending Kummer’s method to all fields, and was overcome by the introduction of compound moduli—for this is the common characteristic of Dedekind’s and Kronecker’s procedure.

51. Normal Fields.—The special properties of a particular field $\Omega$ are closely connected with its relations to the conjugate fields $\Omega ',\Omega '',\dots \Omega ^{(n-1)})$. The most important case is when each of the conjugate fields is identical with $\Omega$: the field is then said to be Galoisian or normal. The aggregate $\mathrm {R} (\Omega ,\Omega ',\dots \Omega ^{(n-1)}))$ of all rational functions of $\theta$ and its conjugates is a normal field: hence every arithmetical field of order $n$ is either normal, or contained in a normal field of a higher order. The roots of an equation $f(\theta )=0$ which defines a normal field are associated with a group of substitutions: if this is Abelian, the field is called Abelian; if it is cyclic, the field is called cyclic. A cyclotomic field is one the elements of which are all expressible as rational functions of roots of unity; in particular the complete cyclotomic field $\mathrm {C} _{m}$, of order $\phi (m)$, is the aggregate of all rational functions of a primitive mth root of unity. To Kronecker is due the very remarkable theorem that all Abelian (including cyclic) fields are cyclotomic: the first published proof of this was given by Weber, and another is due to D. Hubert.

Many important theorems concerning a normal field have been established by Hilbert. He shows that if $\Omega$ is a given normal field of order $m$, and ${\mathfrak {p}}$ any of its prime ideals, there is a finite series of associated fields $\Omega _{1},\Omega _{2},\!\And \!\!\!\!\mathrm {c} .$, of orders $m_{1},m_{2},\!\And \!\!\!\!\mathrm {c} .$, such that $m_{i}\equiv 0{\pmod {m_{i+1}}}$, and that if $r_{i}=m/m_{i}$, ${\mathfrak {p}}^{r_{i}}={\mathfrak {p}}_{i}$, a prime ideal in $\Omega _{i}$. If $\Omega _{l}$ is the last of this series, it is called the field of inertia (Trägheitskörper) for 𝔭: next after this comes another field of still lower order called the resolving field (Zerlegungskörper) for 𝔭, and in this field there is a prime of the first degree, 𝔭_𝑙＋1, such that 𝔭_𝑙＋1＝𝔭^𝒌, where 𝒌＝𝑚 ∕𝑚_𝑙. In the field of inertia 𝔭_𝑙＋1 remains a prime, but becomes of higher degree; in Ω_𝑙—1, which is called the branch-field (Verzweigungskörper) it becomes a power of a prime, and by going on in this way from the resolving field to Ω, we obtain (𝑙＋2) representations for any prime ideal of the resolving field. By means of these theorems, Hilbert finds an expression for the exact power to which a rational prime 𝒑 occurs in the discriminant of Ω, and in other ways the structure of Ω becomes more evident. It may be observed that whem 𝑚 is prime the whole series reduces to Ω and the rational field, and we conclude that every prime ideal in Ω is of the first or 𝑚th degree: this is the case, for instance, when 𝑚＝2, and is one of the reasons why quadratic fields are comparatively so simple in character.

52. Quadratic Fields.—Let 𝑚 be an ordinary integer different from ＋1, and not divisible by any square: then if 𝑥, 𝑦 assume all ordinary rational values the expressions 𝑥＋𝑦√𝑚 are the elements of a field which may be called Ω(√𝑚). It should be observed that √𝑚 means one definite root of 𝑥²—𝑚＝0, it does not matter which: it is convenient, however, to agree that √𝑚 is positive when 𝑚 is positive, and 𝑖√𝑚 is negative when 𝑚 is negative. The principal results relating to Ω will now be stated, and will serve as illustrations of §§ 44-51.

In the notation previously used

𝔬＝[1, 1/2(1＋√𝑚)] or [1, √𝑚]

according as 𝑚≡1 (mod 4) or not. In the first case 𝚫＝𝑚, in the second 𝚫＝4𝑚. The field Ω is normal, and every ideal prime in it is of the first degree.

Let 𝒒 be any odd prime factor of 𝑚; then 𝒒＝𝔮², where 𝔮 is the prime ideal [𝒒, 1/2(𝒒＋√𝑚)] when 𝑚≡1 (mod 4) and in other cases [𝒒, √𝑚]. An odd prime 𝒑 of which 𝑚 is a quadratic residue is the product of two prime ideals 𝔭, 𝔭′, which may be written in the form [𝒑, 1/2(𝒂＋√𝑚)], [𝒑, 1/2(𝒂—√𝑚)] or [𝒑, 𝒂＋√𝑚], [𝒑, 𝒂—√𝑚], according as 𝑚≡1 (mod 4) or not: here 𝒂 is a root of 𝑥²≡𝑚 (mod 𝒑), taken so as to be odd in the first of the two cases. All other rational odd primes are primes in Ω. For the exceptional prime 2 there are four cases to consider: (i.) if 𝑚≡1 (mod 8), then 2＝[2,1/2(1＋√𝑚)]×[2,1/2(1—√𝑚)]. (ii.) If 𝑚≡5 (mod 8), then 2 is prime: (iii.) if 𝑚≡2 (mod 4), 2＝[2,√𝑚]²: (iv.) if 𝑚＝3 (mod≡4), 2＝[2,1＋√m)². Illustrations will be found in § 44 for the case 𝑚＝23.

53. Normal Residues. Genera.—Hilbert has introduced a very convenient definition, and a corresponding symbol, which is a generalization of Legendre’s quadratic character. Let 𝒏, 𝑚 be rational integers, 𝑚 not a square, 𝑤 any rational prime; we write $\left({n,\ m \over w}\right)=+1$ if, to the modulus 𝑤, 𝒏 is congruent to the norm of an integer contained in Ω(√𝑚); in all other cases we put $\left({n,\ m \over w}\right)=-1$. This new symbol obeys a set of laws, among which may be especially noted $\left({n,\ w \over w}\right)=\left({w,\ n \over w}\right)=\left({n \over w}\right)$ and $\left({n,\ m \over w}\right)=+1$, whenever 𝒏, 𝑚 are prime to 𝒑.

Now let 𝒒 ₁, 𝒒 ₂ , . . . 𝒒_𝑡 be the different rational prime factors of the discriminant of Ω(√𝑚); then with any rational integer 𝒂 we may associate the 𝑡 symbols

$\left({a,\ m \over q_{1}}\right),\ \ \left({a,\ m \over q_{2}}\right),\ .\ .\ .\ \left({a,m \over q_{t}}\right)$

and call them the total character of 𝒂 with respect to Ω. This definition may be extended so as to give a total character for every ideal 𝔞 in Ω, as follows. First let Ω be an imaginary field (𝑚＜0); we put 𝒓＝𝑡, 𝒏＝𝖭(𝔞), and call

$\left({{\overline {n}},\ m \over q_{1}}\right),\ .\ .\ .\ \left({{\overline {n}},\ m \over q_{r}}\right)$

the total character of 𝔞. Secondly, let Ω be a real field; we first determine the 𝑡 separate characters of —&#;8198;1, and if they are all positive we put \overline{𝒏}＝ ＋𝖭(𝔞), 𝒓＝𝑡, and adopt the 𝒓 characters just written above as those of 𝔞. Suppose, however, that one of the characters of —1 is negative; without loss of generality we may take it to be that with reference to 𝒒_𝑡. We then put 𝒓＝𝑡—1, 𝒏＝±𝖭(𝔞) taken with such a sign that $\left({{\overline {n}},\ m \over q_{t}}\right)=+1$, and take as the total character of 𝔞 the symbols $\left({{\overline {n}},\ m \over q_{i}}\right)$ for 𝑖＝1, 2, . . . (𝑡 — 1).

With these definitions it can be proved that all ideals of the same class have the same total character, and hence there is a distribution of classes into genera, each genus containing those classes for which the total character is the same (cf. § 36).

Moreover, we have the fundamental theorem that an assigned set of 𝒓 units ±1 corresponds to an actually existing genus if, and only if, their product is ＋1, so that the number of actually existing genera is 2^𝒓—1. This is really equivalent to a theorem about quadratic forms first stated and proved by Gauss; the same may be said about the next proposition, which, in its natural order, is easily proved by the method of ideals, whereas Gauss had to employ the theory of ternary quadratics.

Every class of the principal genus is the square of a class.

An ambiguous ideal in Ω is defined as one which is unaltered by the change of √𝑚 to — √𝑚 (that is, it is the same as its conjugate) and not divisible by any rational integer except ±1. The only ambiguous prime ideals in Ω are those which are factors of its discriminant. Putting 𝚫＝𝔮₁² 𝔮₂² . . . 𝔮_𝑡², there are in Ω exactly 2^𝑡 ambiguous ideals: namely, those factors of 𝚫, including 𝔬, which are not divisible by any square. It is a fundamental theorem, first proved by Gauss, that the number of ambiguous classes is equal to the number of genera.

54. Class-Number.—The number of ideal classes in the field Ω(√𝑚) may be expressed in the following forms:—

(i.) 𝑚＜0:

$h={\tau \over 2\Delta _{n}}\Sigma \left({\Delta \over n}\right)n\qquad (n=1,\,2,\,.\,.\,.,\,-\Delta )$

(ii.) 𝑚＞0:

$h={1 \over 2\log \epsilon }\log {\Pi \sin {b\pi \over \Delta } \over \Pi \sin {a\pi \over \Delta }}$

In the first of these formulae 𝜏 is the number of units contained in Ω; thus 𝜏＝6 for 𝚫＝—3, 𝜏＝4 for 𝚫＝—4, 𝜏＝2 in other cases. In the second formula, 𝜖 is the fundamental unit, and the products are taken for all the numbers of the set (1, 2, . . . 𝚫) for which $\left({\Delta \over a}\right)=+1$ , $\left({\Delta \over b}\right)=-1$ respectively. In the ideal theory the only way in which these formulae have been obtained is by a modification of Dirichlet’s method; to prove them without the use of transcendental analysis would be a substantial advance in the theory.

55. Suppose that any ideal in Ω is expressed in the form [𝜔₁, 𝜔₂]; then any element of it is expressible as 𝑥𝜔₁＋𝑦𝜔₂, where 𝑥, 𝑦 are rational integers, and we shall have 𝖭 (𝑥𝜔₁＋𝑦𝜔₂)＝𝒂𝑥²＋𝒃𝑥𝑦＋𝒄𝑦², where 𝒂, 𝒃, 𝒄 are rational numbers contained in the ideal. If we put 𝑥＝𝛼𝑥′＋𝛽𝑦′, 𝑦＝𝛾𝑥′＋𝛿𝑦′, where 𝛼, 𝛽, 𝛾, 𝛿 are rational numbers such that 𝛼𝛿—𝛽𝛾＝±1, we shall have simultaneously (𝒂, 𝒃, 𝒄) (𝑥, 𝑦)²＝(𝒂′, 𝒃′, 𝒄′) (𝑥′, 𝑦′)² as in § 32 and also

(𝒂′,𝒃′,𝒄′) (𝑥′,𝑦′)²＝𝖭{𝑥′(𝛼𝜔₁＋𝛾𝜔₂)＋𝑦′(𝛽𝜔₁＋𝛿𝜔₂)}＝𝖭(𝑥′𝜔′₁＋𝑦′𝜔′₂),

where [𝜔′₁, 𝜔′₂] is the same ideal as before. Thus all equivalent forms are associated with the same ideal, and the numbers representable by forms of a particular class are precisely those which are norms of numbers belonging to the associated ideal. Hence the class-number for ideals in Ω is also the class-number for a set of quadratic forms; and it can be shown that all these forms have the same determinant 𝚫. Conversely, every class of forms of determinant 𝚫 can be associated with a definite class of ideals in Ω(√𝑚), where 𝑚＝𝚫 or 1/4𝚫 as the case may be. Composition of form-classes exactly corresponds to the multiplication of ideals: hence the complete analogy between the two theories, so long as they are really in contact. There is a corresponding theory of forms in connexion with a field of order 𝒏: the forms are of the order 𝒏, but are only very special forms of that order, because they are algebraically resolvable into the product of linear factors.

56. Complex Quadratic Forms.—Dirichlet, Smith and others, have discussed forms (𝒂, 𝒃, 𝒄) in which the coefficients are complex integers of the form 𝑚＋𝒏𝑖; and Hermite has considered bilinear forms 𝒂𝑥𝑥′＋𝒃𝑥𝑦′＋𝒃′𝑥′𝑦＋𝒄𝑦𝑦′, where 𝑥′, 𝑦′, 𝒃′ are the conjugates of 𝑥, 𝑦, 𝒃 and 𝒂, 𝒄, are real. Ultimately these theories are connected with fields of the fourth order; and of course in the same way we might consider forms (𝒂, 𝒃, 𝒄) with integral coefficients belonging to any given field of order 𝒏: the theory would then be ultimately connected with a field of order 2𝒏.

57. Kronecker's Method.—In practice it is found convenient to combine the method of Dedekind with that of Kronecker, the main principles of which are as follows. Let 𝖥( 𝑥, 𝑦, 𝑧, . . .) be a polynomial in any number of indeterminates (umbrae, as Sylvester calls them) with ordinary integral coefficients; if 𝒏 is the greatest common measure of the coefficients, we have 𝖥＝𝒏𝖤, where 𝖤 is a primary or unit form. The positive integer 𝒏 is called the divisor of 𝖥; and the divisor of the product of two forms is equal to the product of the divisors of the factors. Next suppose that the coefficients of 𝖥 are integers in a field Ω of order 𝒏. Denoting the conjugate forms by 𝖥′, 𝖥″, . . . 𝖥^(𝒏－1), the product 𝖥𝖥′𝖥″ . . . 𝖥^(𝒏－1)＝𝑓𝖤, where 𝑓 is a real positive integer, and 𝖤 a unit form with real integral coefficients. The natural number 𝑓 is called the norm of 𝖥. If 𝖥, 𝖦 are any two forms (in Ω) we have 𝖭(𝖥𝖦)＝𝖭(𝖥)𝖭(𝖦). Let the coefficients of 𝖥 be 𝛼₁, 𝛼₂, those of 𝖦 𝛽₁, 𝛽₂, &c., and those of 𝖥𝖦 𝛾₁, 𝛾₂, &c.; and let 𝔭 be any prime ideal in Ω. Then if 𝔭^𝑚 is the highest power of 𝔭 contained in each of the coefficients 𝛼_𝑖, and 𝔭^𝒏 the highest power of 𝔭 contained in each of the coefficients 𝛽_𝑖, 𝔭^𝑚＋𝒏 is the highest power of 𝔭 contained by the whole set of coefficients 𝛾_𝑖. Writing dv(𝛼₁, 𝛼₂, . . .) for the highest ideal divisor of 𝛼₁, 𝛼₂, &c., this is called the content of 𝖥; and we have the theorem that the product of the contents of two forms is equal to the content of the product of the forms. Every form is associated with a definite ideal 𝔪, and we have 𝖭(𝖥)＝𝖭(𝔪) if 𝔪 is the content of 𝖥, and 𝖭(𝔪) has the meaning already assigned to it. On the other hand, to a given ideal correspond an indefinite number of forms of which it is the content; for instance (§ 46, end) we can find forms 𝛼𝑥＋𝛽𝑦 of which any given ideal is the content.

58. Now let 𝜔₁, 𝜔₂, . . . 𝜔_𝒏, be a basis of 𝔬; 𝑢₁, 𝑢₂, . . . 𝑢_𝒏 a set of indeterminates; and

𝜉＝𝜔₁𝑢₁＋𝜔₂𝑢₂＋ . . . ＋𝜔_𝒏𝑢_𝒏:

𝜉 is called the fundamental form of Ω. It satisfies the equation 𝖭(𝑥－𝜉)＝0, or

𝖥(𝑥)＝𝑥^𝒏＋𝖴₁𝑥^𝒏－1＋ . . . ＋𝖴_𝒏＝0

where 𝖴₁, 𝖴₂, . . . 𝖴_𝒏 are rational polynomials in 𝑢₁, 𝑢₂, . . .𝑢_𝒏 with rational integral coefficients. This is is called the fundamental equation.

Suppose now that 𝑝 is a rational prime, and that 𝑝＝𝔭^𝒂𝔮^𝒃𝔯^𝒄 where 𝔭, 𝔮, 𝔯, . . . &c., are the different ideal prime factors of 𝑝, then if 𝖥(𝑥) is the left-hand side of the fundamental equation there is an identical congruence

𝖥(𝑥)＝{𝖯(𝑥)}^𝒂{𝖰(𝑥)}^𝒃{𝖱(𝑥)}^𝒄 . . . (mod 𝑝)

where 𝖯(𝑥), 𝖰(𝑥), &c., are prime functions with respect to 𝔭. The meaning of this is that if we expand the expression on the right-hand side of the congruence, the coefficient of every term 𝑥^𝑙𝑢₁^𝑚. . . 𝑢_𝒏^𝑡 will be congruent, mod 𝑝, to the corresponding coefficient in 𝖥(𝑥). If 𝑓, 𝑔, ℎ, &c., are the degrees of 𝔭, 𝔮, 𝔯, &c. (§ 47), then 𝑓, 𝑔, ℎ, . . . are the dimensions in 𝑥, 𝑢₁, 𝑢₂,. . . 𝑢_𝒏 of the forms of 𝖯, 𝖰, 𝖱, respectively. For every prime 𝑝, which is not a factor of 𝚫, 𝒂＝𝒃＝𝒄＝. . .＝1 and 𝖥(𝑥) is congruent to the product of a set of different prime factors, as many in number as there are different ideal prime factors of 𝑝. In particular, if 𝑝 is a prime in Ω, 𝖥(𝑥) is a prime function (mod 𝑝) and conversely.

It generally happens that rational integral values 𝒂₁, 𝒂₂, . . . 𝒂_𝒏 can be assigned to 𝑢₁, 𝑢₂, . . . 𝑢_𝒏 such that 𝖴_𝒏, the last term in the fundamental equation, then has a value which is prime to 𝑝. Supposing that this condition is satisfied, let 𝒂₁𝜔₁＋𝒂₂𝜔₂＋ . . .＋𝒂_𝒏𝜔_𝒏＝𝛼; and let 𝖯₁(𝛼) be the result of putting 𝑥＝𝛼, 𝑢_𝑖＝𝒂_𝑖 in 𝖯(𝑥). Then the ideal 𝔭 is completely determined as the greatest common divisor of 𝑝 and 𝖯₁(𝛼); and similarly for the other prime factors of 𝑝. There are, however, exceptional cases when the condition above stated is not satisfied.

59. Cyclotomy.—It follows from de Moivre’s theorem that the arithmetical solution of the equation 𝑥^𝑚－1＝0 corresponds to the division of the circumference of a circle into 𝑚 equal parts. The case when 𝑚 is composite is easily made to depend on that where 𝑚 is a power of a prime; if 𝑚 is a power of 2, the solution is effected by a chain of quadratic equations, and it only remains to consider the case when 𝑚＝𝑞^𝜅, a power of an odd prime. It will be convenient to write 𝜇＝𝜙(𝑚)＝𝑞^𝜅－1(𝑞－1); if we also put 𝑟＝𝑒^{2𝜋𝑖∕𝑚}, the primitive roots of 𝑥^𝑚＝1 will be 𝜇 in number, and represented by 𝑟, 𝑟^𝒂, 𝑟^𝒃, &c. where 1, 𝒂, 𝒃, &c., form a complete set of prime residues to the modulus 𝑚. These will be the roots of an irreducible equation 𝑓(𝑥)＝0 of degree 𝜇; the symbol 𝑓(𝑥) denoting (𝑥^𝑚－1)÷(𝑥^𝑚/𝑞－1). There are primitive roots of the congruence 𝑥^𝜇＝1 (mod 𝑚); let 𝑔 be any one of these. Then if we put 𝑟^𝑔ℎ＝𝑟_ℎ, we obtain all the roots of 𝑓(𝑥)＝0 in a definite cyclical order (𝑟₁, 𝑟₂ . . .𝑟_𝜇; and the change of 𝑟 into 𝑟^𝑔 produces a cyclical permutation of the roots. It follows from this that every cyclic polynomial in 𝑟₁, 𝑟₂ . . .𝑟_𝜇 with rational coefficients is equal to a rational number. Thus if we write 𝑙＋𝒂𝑔＋𝒃𝑔²＋.＋𝑘𝑔^𝜇－1＝𝒏, we have, in virtue of 𝑟_ℎ＝𝑟^𝑔𝑘, 𝑟₁^𝒂𝑟₂^𝒃𝑟_𝜇－1^𝑘𝑟_𝜇^𝑙＝𝑟^𝒏, and, if we use 𝖲 to denote cyclical summation, 𝖲(𝑟₁^𝒂𝑟₂^𝒃. . .𝑟_𝜇^𝑙)＝𝑟^𝒏＋𝑟^𝒏𝑔＋ . . . ＋𝑟^{𝒏𝑔𝜇－1}, the sum of the 𝒏th powers of all the roots of 𝑓(𝑥)＝0, and this is a rational integer or zero. Since every cyclic polynomial is the sum of parts similar to 𝖲(𝑟₁^𝒂𝑟₂^𝒃. . .𝑟_𝜇^𝑙), the theorem is proved. Now let 𝑒, 𝑓 be any two conjugate factors of 𝜇, so that 𝑒𝑓＝𝜇, and let

𝜂_𝑖＝𝑟_𝑖＋𝑟_𝑖＋𝑒＋𝑟_𝑖＋2𝑒＋. . . ＋𝑟_{𝑖＋(𝑓－1)𝑒}       (𝑖＝1, 2,. . .𝑒)

then the elementary symmetric functions 𝚺𝜂_𝑖, 𝚺𝜂_𝑖𝜂^𝑗, &c., are cyclical functions of the roots of 𝑓(𝑥)＝0 and therefore have rational values which can be calculated: consequently 𝜂₁, 𝜂₂, . . .𝜂_𝑒, which are called the 𝑓-nomial periods, are the roots of an equation 𝖥(𝜂)＝𝜂^𝑒＋𝒄₁𝜂^𝑒－1＋ . . . ＋𝒄_𝑒＝0 with rational integral coefficients. This is irreducible, and defines a field of order 𝑒 contained in the field defined by 𝑓(𝑥)＝0. Moreover, the change of 𝑟 into 𝑟^𝑔 alters 𝒏_𝑖 into 𝜂_𝑖＋1, and we have the theorem that any cyclical function of 𝜂₁, 𝜂₂, . . . 𝒏_𝑒 is rational. Now let ℎ, 𝑘 be any conjugate factors of 𝑓 and put

𝑧_𝑖＝𝑟_𝑖＋𝑟_{𝑖＋ℎ𝑒}＋𝑟^𝑖_＋2ℎ𝑒＋ . . . 𝑟^𝑖_{＋(𝑓－ℎ)𝑒}      (𝑖＝1, 2, 3,)

then 𝜁₁,𝜁_1＋𝑒,𝜁_1＋2𝑒. . .𝜁_{1＋(ℎ－1)𝑒} will be the roots of an equation

𝖦(𝜁)＝𝜁^ℎ－𝜂₁𝜁^ℎ－1＋𝒄₂𝜁^ℎ－2＋ . . . ＋𝒄_ℎ＝0

the coefficients of which are expressible as rational polynomials in 𝜂₁. Dividing ℎ into two conjugate factors, we can deduce from 𝖦(𝜁)＝0 another period equation, the coefficients of which are rational polynomials in 𝜂₂, 𝜁₁, and so on. By choosing for 𝑒, ℎ, &c., the successive prime factors of 𝜇, ending up with 2, we obtain a set of equations of prime degree, each rational in the roots of the preceding equations, and the last having 𝑟₁ and 𝑟₁^－1 for its roots. Thus to take a very interesting historical case, let 𝑚＝17, so that 𝜇＝16＝2⁴, the equations are all quadratics, and if we take 3 as the primitive root of 17, they are

𝜂²＋𝜂－4＝0,        𝜁²－𝜂𝜁－1＝0

2𝜆²－2𝜁𝜆＋(𝜂𝜁－𝜂＋𝜁－3)＝0,   𝜌²－𝜆𝜌＋1＝0.

If two quantities (real or complex) 𝒂 and 𝒃 are represented in the usual way by points in a plane, the roots of 𝑥²＋𝒂𝑥＋𝒃＝0 will be represented by two points which can be found by a Euclidean construction, that is to say, one requiring only the use of rule and compass. Hence a regular polygon of seventeen sides can be inscribed in a given circle by means of a Euclidean construction; a fact first discovered by Gauss, who also found the general law, which is that a regular polygon of 𝑚 sides can be inscribed in a circle by Euclidean construction if and only if 𝜙(𝑚) is a power of 2; in other words 𝑚＝2^𝜅𝖯 where 𝖯 is a product of different odd primes, each of which is of the form 2^𝒏＋1.

Returning to the case 𝑚＝𝑞^𝜅, we shall call the chain of equations 𝖥(𝜂)＝0, &c., when each is of prime degree, a set of Galoisian auxiliaries. We can find different sets, because in forming them we can take the prime factors of 𝜇 in any order we like; but their number is always the same, and their degrees always form the same aggregate, namely, the prime factors of 𝜇. No other chain of auxiliaries having similar properties can be formed containing fewer equations of a given prime degree 𝑝; a fact first stated by Gauss, to whom this theory is mainly due. Thus if 𝑚＝𝑞^𝜅 we must have at least (𝜅－1) auxiliaries of order 𝑞, and if 𝑞－1＝2^𝛼𝑝^𝛽 . . ., we must also have 𝛼 quadratics, 𝛽 equations of order 𝑝, an so on. For this reason a set of Galoisian auxiliaries may be regarded as providing the simplest solution of the equation 𝑓(𝑥)＝0.

60. When 𝑚 is an odd prime 𝑝, there is another very interesting way of solving the equation (𝑥^𝑝－1)÷(𝑥－1)＝0. As before let (𝑟₁, 𝑟₂, . . . 𝑟_𝑝－1) be its roots arranged in a cycle by means of a primitive root of 𝑥^𝑝－1≡1 (mod 𝑝); and let 𝜖 be a primitive root of 𝜖^𝑝－1＝1. Also let

𝜃₁＝𝑟₁＋𝜖𝑟₂＋𝜖²𝑟₃＋ . . . ＋𝜖𝑝－2𝑟𝑝－1
𝜃𝑘＝𝑟₁＋𝜖𝑘𝑟₂＋𝜖2𝑘𝑟₃＋ . . . ＋𝜖－𝑘𝑟𝑝－1	(𝑘＝2, 3, . . . 𝑝－2)

so that 𝜃_𝑘 is derived from 𝜃₁ by changing 𝜖 into 𝜖^𝑘.

The cyclical permutation (𝑟₁, 𝑟₂, . . .𝑟_𝑝－1) applied to 𝜃_𝑘 converts it into 𝜖^－𝑘𝜃_𝑘; hence 𝜃₁𝜃_𝑘/𝜃_𝑘＋1 is unaltered, and may be expressed as a rational, and therefore as an integral function of 𝜖. It is found by calculation that we may put

$\psi _{k}(\epsilon )={\theta _{1}\theta _{k} \over \theta _{k+1}}=\sum _{m=2}^{m=p+1}\epsilon ^{{\mbox{ind}}\ m+k\,{\mbox{ind}}(p+1-m)}\qquad [k=1,\ 2,\,.\,.\,.\,(p-3)]$

while

𝜃₁𝜃_𝑝－2＝－𝑝.

In the exponents of 𝜓_𝑘(𝜖) the indices are taken to the base 𝑔 used to establish the cyclical order (𝑟₁, 𝑟₂, . . . 𝑟_𝑝－1). Multiplying together the (𝑝－2) preceding equalities, the result is

𝜃₁^𝑝－1＝－𝑝𝜓₁(𝜖)𝜓₂(𝜖) . . . 𝜓_𝑝－3(𝜖)＝𝖱(𝜖)

where 𝖱(𝜖) is a rational integral function of 𝜖 the degree of which, in its reduced form, is less than 𝜙(𝑝－1). Let 𝜌 be any one definite root of 𝑥^𝑝－1＝𝖱(𝜖), and put 𝜃₁＝𝜌: then since

${\theta _{1}^{k} \over \theta _{k}}=\psi _{1}\psi _{2}...\psi _{k-1}$

we must take 𝜃_𝑘＝𝜌^𝑘/𝜓₁𝜓₂ . . . 𝜓_𝑘－1＝𝖱_𝑘(𝜖)𝜌^𝑘 , where 𝖱_𝑘(𝜖) is a rational function of 𝜖, which we may suppose put into its reduced integral form; and finally, by addition of the equations which define 𝜃₁, 𝜃₂, &c.,

(𝑝－1)𝑟₁＝𝜌＋𝖱₂(𝜖)𝜌²＋𝖱₃(𝜖)𝜌³＋ . . . ＋𝖱_𝑝－2(𝜖)𝜌^𝑝－2.

If in this formula we change 𝜌 into 𝜖^－ℎ𝜌, and 𝑟₁ into 𝑟_ℎ＋1, it still remains true.

It will be observed that this second mode of solution employs a Lagrangian resolvent 𝜃₁; considered merely as a solution it is neither so direct nor so fundamental as that of Gauss. But the form of the solution is very interesting; and the auxiliary numbers 𝜓(𝜖) have many curious properties, which have been investigated by Jacobi, Cauchy and Kronecker.

61. When 𝑚＝𝑞^𝜅, the discriminant of the corresponding cyclotomic field is ±𝑞^𝜆, where 𝜆＝𝑞^𝜅－1(𝜅𝑞－𝜅－1). The prime 𝑞 is equal to 𝔮^𝜇, where 𝜇＝𝜙(𝑚)＝𝑞^𝜅－1(𝑞－1), and 𝔮 is a prime ideal of the first degree. If 𝑝 is any rational prime distinct from 𝑔, and 𝑓 the least exponent such that 𝑝^𝑓≡1 (mod. 𝑚), 𝑓 will be a factor of 𝜇, and putting 𝜇/𝑓＝𝑒, we have 𝑝＝𝔭₁𝔭₂ . . .𝔭_𝑒, where 𝔭₁, 𝔭₂ . . . 𝔭_𝑒 are different prime ideals each of the 𝑓th degree. There are similar theorems for the case when 𝑚 is divisible by more than one rational prime.

Kummer has stated and proved laws of reciprocity for quadratic and higher residues in what are called regular fields, the definition of which is as follows. Let the field be 𝖱(𝑒^{2𝜋𝑖/𝑝}), where 𝑝 is an odd prime; then this field is regular, and 𝑝 is said to be a regular prime, when ℎ, the number of ideal classes in the field, is not divisible by 𝑝. Kummer proved the very curious fact that 𝑝 is regular if, and only if, it is not a factor of the denominators of the first 1/2(𝑝－3) Bernoullian numbers. He also succeeded in showing that in the field 𝖱(𝑒^{2𝜋𝑖/𝑝}) the equation 𝛼^𝑝＋𝛽^𝑝＋𝛾^𝑝＝0 has no integral solutions whenever ℎ is not divisible by 𝑝². What is known as the “last” theorem of Fermat is his assertion that if 𝑚 is any natural number exceeding 2, the equation 𝑥^𝑚＋𝑦^𝑚＝𝑧^𝑚 has no rational solutions, except the obvious ones for which 𝑥𝑦𝑧＝0. It would be sufficient to prove Fermat’s theorem for all prime values of 𝑚; and whenever Kummer’s theorem last quoted applies, Fermat’s theorem will hold. Fermat’s theorem is true for all values of 𝑚 such that 2＜𝑚＜101, but no complete proof of it has yet been obtained.

Hilbert has studied in considerable detail what he calls Kummer fields, which are obtained by taking 𝑥, a primitive 𝑝th root of unity, and 𝑦 any root of 𝑦^𝑝－𝑎＝0, where 𝑎 is any number in the field 𝖱(𝑥) which is not a perfect 𝑝th power in that field. The Kummer field is then 𝖱(𝑥, 𝑦), Consisting of all rational functions of 𝑥 and 𝑦. Other fields that have been discussed more or less are general cubic fields, some special biquadratic and a few Abelian fields not cyclic.

Among the applications of cyclotomy may be mentioned the proof which it affords of the theorem, first proved by Dirichlet, that if 𝑚, 𝑛 are any two rational integers prime to each other, the linear form 𝑚𝑥＋𝑛 is capable of representing an infinite number of primes.

62. Gauss’s Sums.—Let 𝑚 be any positive real integer; then

$\sum _{s=0}^{s=m-1}e^{2s^{2}\pi i/m}={\frac {i+i(-i)^{m}}{1+i}}{\sqrt {m}}$

This remarkable formula, when 𝑚 is prime, contains results which were first obtained by Gauss, and thence known as Gauss’s sums. The easiest method of proof is Kronecker’s, which consists in finding the value of ∫{𝑒^{2𝜋𝑖𝑧²/𝑚}𝑑𝑧/(1－𝑒^{2𝜋𝑖𝑧})}, taken round an appropriate contour. It will be noticed that one result of the formula is that the square root of any integer can be expressed as a rational function of roots of unity.

The most important application of the formula is the deduction from it of the law of quadratic reciprocity for real primes: this was done by Gauss.

63. One example may be given of some remarkable formulae giving explicit solutions of representations of numbers by certain quadratic forms. Let 𝑝 be any odd prime of the form 7𝑛＋2; then we shall have 𝑝＝7𝑛＋2＝𝑥²＋7𝑦², where 𝑥 is determined by the congruences

2𝑥≡(3𝑛)!/(𝑛)! (2𝑛)!(mod 𝑝); 𝑥≡3 (mod 7).

This formula was obtained by Eisenstein, who proved it by investigating properties of integers in the field generated by 𝜂³－21𝜂－7＝0, which is a component of the field generated by seventh roots of unity. The first formula of this kind was given by Gauss, and relates to the case 𝑝＝4𝑛＋1＝𝑥²＋𝑦²; he conceals its connexion with complex numbers. Probably there are many others which have not yet been stated.

64. Higher Congruences. Functional Moduli.—Suppose that 𝑝 is a real prime, and that 𝑓(𝑥), 𝜙(𝑥) are polynomials in 𝑥 with rational integral coefficients. The congruence 𝑓(𝑥)≡𝜙(𝑥) (mod 𝑝) is identical when each coefficient of 𝑓 is congruent, mod 𝑝, to the corresponding coefficient of 𝜙. It will be convenient to write, under these circumstances, 𝑓∼𝜙(mod 𝑝) and to say that 𝑓, 𝜙 are equivalent, mod 𝑝. Every polynomial of degree ℎ is equivalent to another of equal or lower degree, which has none of its coefficients negative, and each of them less than 𝑝. Such a polynomial, with unity for the coefficient of the highest power of 𝑥 contained in it, may be called a reduced polynomial with respect to 𝑝. There are, in all, 𝑝^ℎ reduced polynomials of degree ℎ. A polynomial may or may not be equivalent to the product of two others of lower degree than itself; in the latter case it is said to be prime. In every case, 𝖥 being any polynomial, there is an equivalence 𝖥∼𝑐𝑓₁𝑓₂ . . . 𝑓_𝑙 where 𝑐 is an integer and 𝑓₁, 𝑓₂,...𝑓_𝑙 are prime functions; this resolution is unique. Moreover, it follows from Fermat’s theorem that {𝖥(𝑥)}^𝑝∼𝖥(𝑥^𝑝),{𝖥(𝑥)}^𝑝²∼𝖥(𝑥^𝑝²), and so on.

As in the case of equations, it may be proved that, when the modulus is prime, a congruence 𝑓(𝑥)≡0 (mod 𝑝) cannot have more in congruent roots than the index of the highest power of 𝑥 in 𝑓(𝑥), and that if 𝑥≡𝜉 is a solution, 𝑓(𝑥)∼(𝑥－𝜉)𝑓₁(𝑥), where 𝑓₁(𝑥) is another polynomial. The solutions of 𝑥^𝑝≡𝑥 are all the residues of 𝑝; hence 𝑥^𝑝－𝑥∼𝑥(𝑥＋1)(𝑥＋2) . . .(𝑥＋𝑝－1), where the right-hand expression is the product of all the linear functions which are prime to 𝑝. A generalization of this is contained in the formula

𝑥(𝑥^𝑝𝑚－1－1)∼𝚷𝑓(𝑥) (mod 𝑝)

where the product includes every prime function 𝑓(𝑥) of which the degree is a factor of 𝑚. By a process similar to that employed in finding the equation satisfied by primitive 𝑚th roots of unity, we can find an expression for the product of all prime functions of a given degree 𝑚, and prove that their number is (𝑚＞1)

1/𝑚(𝑝^𝑚－𝚺𝑝^𝑚/𝑎＋𝚺𝑝^{𝑚/𝑎𝑏}－. . .)

where 𝒂, 𝑏, 𝑐 . . . are the different prime factors of 𝑚. Moreover, if 𝖥 is any given function, we can find a resolution

𝖥∼𝑐𝖥₁𝖥₂ . . . 𝖥_𝑚(mod 𝑝)

where 𝑐 is numerical, 𝖥₁ is the product of all prime linear functions which divide 𝖥, 𝖥₂ is the product of all the prime quadratic factors, and so on.

65. By the functional congruence 𝜙(𝑥)≡𝜓(𝑥) (mod 𝑝,𝑓(𝑥)) is meant that polynomials 𝖴, 𝖵 can be found such that 𝜙(𝑥)＝𝜓(𝑥)＋𝑝𝖴＋𝖵𝑓(𝑥) identically. We might also write 𝜙(𝑥)∼𝜓(𝑥) (mod 𝑝, 𝑓(𝑥)); but this is not so necessary here as in the preceding case of a simple modulus. Let 𝑚 be the degree of 𝑓(𝑥); without loss of generality we may suppose that the coefficient of 𝑥^𝑚 is unity, and it will be further assumed that 𝑓(𝑥) is a prime function, mod 𝑝. Whatever the dimensions of 𝜙(𝑥), there will be definite functions 𝜒(𝑥), 𝜙₁(𝑥) such that 𝜙(𝑥)＝𝑓(𝑥)𝜒(𝑥)＋𝜙₁(𝑥) where 𝜙₁(𝑥) is of lower dimension than 𝑓(𝑥); moreover, we may suppose 𝜙₁(𝑥) replaced by the equivalent reduced function 𝜙₂(𝑥) mod 𝑝. Finally then, 𝜙≡𝜙₂ (mod 𝑝, 𝑓(𝑥)) where 𝜙₂ is a reduced function, mod 𝑝, of order not greater than (𝑚－1). If we put 𝑝^𝑚＝𝑛, there will be in all (including zero) 𝑛 residues to the compound modulus (𝑝, 𝑓): let us denote these by 𝖱₁, 𝖱₂, . . . 𝖱_𝑛. Then (cf. § 28) if we reject the one zero residue (𝖱_𝑛, suppose) and take any function 𝜙 of which the residue is not zero, the residues of 𝜙𝖱₁, 𝜙𝖱₂, . . . 𝜙𝖱_𝑛－1 will all be different, and we conclude that 𝜙^𝑛－1≡1 (mod 𝑝, 𝑓). Every function therefore satisfies 𝜙^𝑛∼𝜙 (mod 𝑝, 𝑓); by putting 𝜙＝𝑥 we obtain the principal theorem stated in § 64.

A still more comprehensive theory of compound moduli is due to Kronecker; it will be sufficiently illustrated by a particular case. Let 𝑚 be a fixed natural number; 𝖷, 𝖸, 𝖹, 𝖳 assigned polynomials, with rational integral coefficients, in the independent variables 𝑥, 𝑦, 𝑧; and let 𝖴 be any polynomial of the same nature as 𝖷, 𝖸, 𝖹, 𝖳. We may write 𝖴∼0 (mod 𝑚, 𝖷, 𝖸, 𝖹, 𝖳) to express the fact that there are integral polynomials 𝖬, 𝖷′, 𝖸′, 𝖹′, 𝖳′ such that

𝖴＝𝑚𝖬＋𝖷′𝖷＋𝖸′𝖸＋𝖹′𝖹＋𝖳′𝖳

identically. In this notation 𝖴∼𝖵 means that 𝖴－𝖵∼0. The number of independent variables and the number of functions in the modulus are unrestricted; there may be no number 𝑚 in the modulus, and there need not be more than one. This theory of Kronecker’s is admirably adapted for the discussion of all algebraic problems of an arithmetical character, and is certain to attain a high degree of development.

It is worth mentioning that one of Gauss’s proofs of the law of quadratic reciprocity (Gött. Nachr. 1818) involves the principle of a compound modulus.

66. Forms of Higher Degree:—Except for the case alluded to at the end of § 55, the theory of forms of the third and higher degree is still quite fragmentary. C. Jordan has proved that the class number is finite. H. Poincaré has discussed the classification of ternary and quaternary cubics. With regard to the ternary cubic it is known that from any rational solution of 𝑓＝0 we can deduce another by a process which is equivalent to finding the tangential of a point (𝑥₁, 𝑦₁, 𝑧₁) on the curve, that is, the point where the tangent at (𝑥₁, 𝑦₁, 𝑧₁) meets the curve again. We thus obtain a series of solutions (𝑥₁, 𝑦₁, 𝑧₁), (𝑥₂, 𝑦₂, 𝑧₂), &c., which may or may not be periodic. E. Lucas and J. J. Sylvester have proved that for certain cubics 𝑓＝0 has no rational solutions; for instance 𝑥³＋𝑦³－𝖠𝑧³＝0 has rational solutions only if 𝖠＝𝑎𝑏(𝑎＋𝑏)/𝒄³, where 𝑎, 𝑏, 𝑐 are rational integers. Waring asserted that every natural number can be expressed as the sum of not more than 9 cubes, and also as the sum of not more than 19 fourth powers; these propositions have been neither proved nor disproved.

67. Results derived from Elliptic and Theta Functions.—For the sake of reference it will be convenient to give the expressions for the four Jacobian theta functions. Let 𝜔 be any complex quantity such that the real part of 𝑖𝜔 is negative; and let 𝑞＝𝑒^𝜋𝑖𝜔. Then

$\theta _{00}(v)=\sum _{-\infty }^{\infty }1+q^{s^{2}}e^{2s\pi iv}=1+2q\cos 2\pi v+2q^{4}\cos 4\pi v+2q^{9}\cos 6\pi v+...$

$\qquad \quad =\prod _{1}^{\infty }(1-q^{2s})(1+2q^{2s-1}\cos 2\pi v+q^{4s-2}),$

$\theta _{01}(v)=1-2q\cos 2\pi v+2q^{4}\cos 4\pi v-2q^{9}\cos 6\pi v+...$

$\qquad \quad =\prod _{1}^{\infty }(1-q^{2s})(1-2q^{2s-1}\cos 2\pi v+q^{4s-2}),$

$\theta _{10}(v)=2q^{1 \over 4}\cos \pi v+2q^{9 \over 4}\cos 3\pi v+2q^{25 \over 4}\cos 5\pi v+...$

$\qquad \quad =2q^{1 \over 4}\cos \pi v\prod _{1}^{\infty }(1-q^{2s})(1+2q^{2s}\cos 2\pi v+q^{4s})$

$\theta _{11}(v)=2q^{1 \over 4}\sin \pi v-2q^{9 \over 4}\sin 3\pi v+2q^{25 \over 4}\sin 5\pi v-...$

$\qquad \quad =2q^{1 \over 4}\sin \pi v\prod _{1}^{\infty }(1-q^{2s})(1-2q^{2s}\cos 2\pi v+q^{4s})$

Instead of 𝜃₀₀(0), &c., we write 𝜃₀₀, &c. Clearly 𝜃₁₁＝0; we have the important identities

𝜃₁₁′＝𝜋𝜃₀₀𝜃₁₀𝜃₀₁    𝜃₀₀⁴＝𝜃₀₁⁴＋𝜃₁₀⁴

where 𝜃₁₁′ means the value of 𝑑𝜃₁₁(𝑣)/𝑑𝑣 for 𝑣＝0. If, now, we put

${\sqrt {\kappa }}={\theta _{10} \over \theta _{00}},\quad {\sqrt {\kappa '}}={\theta _{01} \over \theta _{00}},\quad u=\pi \theta _{00}{}^{2}v$

so that 𝑘²＋𝑘′²＝1, we shall have

${\theta _{11}(v) \over \theta _{01}(v)}={\sqrt {\kappa }}\cdot {\mbox{sn}}\ u,\ \ {\theta _{10}(v) \over \theta _{01}(v)}={\sqrt {\kappa \over \kappa '}}\cdot {\mbox{cn}}\ u,\;{\theta _{00}(v) \over \theta _{01}(v)}={1 \over {\sqrt {\kappa '}}}\cdot {\mbox{dn}}\ u$

and, supposing for simplicity that 𝑖𝜔 is a real negative quantity,

𝜋𝜃₀₀²＝2𝖪,    𝜔𝜋𝜃₀₀²＝2𝑖𝖪′,    𝜔＝𝑖𝖪′/𝖪,

the notation being that which is now usual for the elliptic functions. It is found that

${\begin{matrix}{\kappa {\mbox{K}} \over \pi }{\mbox{sn}}\;2{\mbox{K}}u&=&2\sum _{1}^{\infty }{q^{s-{1 \over 2}} \over 1-q^{2s-1}}\sin \;(2s-1)\;\pi u,\\{\kappa {\mbox{K}} \over \pi }{\mbox{cn}}\;2{\mbox{K}}u&=&2\sum _{1}^{\infty }{q^{s-{1 \over 2}} \over 1-q^{2s-1}}\cos \;(2s-1)\;\pi u,\\{{\mbox{K}} \over \pi }{\mbox{dn}}\;2{\mbox{K}}u&=&{1 \over 2}+2\sum _{1}^{\infty }{q^{s} \over 1+q^{2s}}\cos \;2s\pi u.\end{matrix}}$

From the last formula, by putting 𝑢＝0, we obtain

$1+4\sum _{1}^{\infty }{q^{s} \over 1+q^{2s}}={2{\mbox{K}} \over \pi }=\theta _{00}{}^{2}(1+2q+2q^{4}+2q^{9}+...)^{2}$,

and hence, by expanding both sides in ascending powers of 𝑞, and equating the coefficients of 𝑞^𝑛, we arrive at a formula for the number of ways of expressing 𝑛 as the sum of two squares. If 𝛿 is any odd divisor of 𝑛, including 1 and 𝑛 itself if 𝑛 is odd, we find as the coefficient of 𝑞^𝑛 in the expansion of the left-hand side 4𝚺(－1)^{1/2(𝛿－1)}; on the right-hand side the coefficient enumerates all the solutions 𝑛＝(±𝑥)²＋(±𝑦)², taking account of the different signs (except for 0²) and of the order in which the terms are written (except when 𝑥²＝𝑦²). Thus if 𝑛 is an odd prime of the form 4𝑘＋1, 𝚺(－1)^{1/2(𝛿－1)}＝2, and the coefficient of 𝑞^𝑛 is 8, which is right, because the one possible composition 𝑛＝𝑎²＋𝑏² may be written 𝑛＝(±𝑎)²＋(±𝑏)²＝(±𝑏)²＋(±𝑎)², giving eight representations.

By methods of a similar character formulae can be found for the number of representations of a number as the sum of 4, 6, 8 squares respectively. The four-square theorem has been stated in § 41; the eight-square theorem is that the number of representations of a number as the sum of eight squares is sixteen times the sum of the cubes of its factors, if the given number is odd, while for an even number it is sixteen times the excess of the cubes of the even factors above the cubes of the odd factors. The five-square and seven-square theorems have not been derived from 𝑞-series, but from the general theory of quadratic forms.

68. Still more remarkable results are deducible from the theory of the transformation of the theta functions. The elementary formulae are

${\begin{matrix}\theta _{11}(u,\;\omega +1)=e^{\pi i/4}\theta _{11}(u,\;\omega ),&\theta _{10}(u,\;\omega +1)=e^{\pi i/4}\theta _{10}(u,\;\omega )\\\theta _{01}(u,\;\omega +1)=\theta _{00}(u,\;\omega ),&\theta _{00}(u,\;\omega +1)=\theta _{01}(u,\;\omega )\end{matrix}}$

${\begin{matrix}e^{-\pi iu^{2}/\omega }\theta _{11}\left({u \over \omega },-{1 \over \omega }\right)&=&-i{\sqrt {-i\omega }}\theta _{11}(u,\;\omega ),\\e^{-\pi iu^{2}/\omega }\theta _{10}\left({u \over \omega },-{1 \over \omega }\right)&=&{\sqrt {-i\omega }}\theta _{10}(u,\;\omega ),\\e^{-\pi iu^{2}/\omega }\theta _{10}\left({u \over \omega },-{1 \over \omega }\right)&=&-i{\sqrt {-i\omega }}\theta _{01}(u,\;\omega ),\\e^{-\pi iu^{2}/\omega }\theta _{00}\left({u \over \omega },-{1 \over \omega }\right)&=&-i{\sqrt {-i\omega }}\theta _{00}(u,\;\omega ),\end{matrix}}$

where √－𝑖𝜔 is to be taken in such a way that its real part is positive. Taking the definition of 𝜅 given in § 67, and considering 𝜅 as a function of 𝜔, we find

𝜅(𝜔＋1)＝𝑖𝜃₁₀²/𝜃₀₁²＝𝑖𝜅(𝜔)/𝜅′(𝜔)

$\kappa \left(-{1 \over \omega }\right)=\theta _{01}{}^{2}/\theta _{00}{}^{2}=\kappa '(\omega )$.

For convenience let 𝜅²(𝜔)＝𝜎: then the substitutions (𝜔,𝜔＋1) and (𝜔,－𝜔^－1) convert 𝜎 into 𝜎/(𝜎－1) and (1－𝜎) respectively. Now if 𝛼, 𝛽, 𝛾, 𝛿 are any real integers such that 𝛼𝛿－𝛽𝛾＝1, the substitution [𝜔,(𝛼𝜔＋𝛽)/(𝛾𝜔＋𝛿)] can be compounded of (𝜔, 𝜔＋1) and (𝜔,－𝜔^－1); the effect on 𝜎 will be the same as if we apply a corresponding substitution compounded of [𝜎, 𝜎/(𝜎－1)] and [𝜎, 1－𝜎]. But these are periodic and of order 3, 2 respectively; therefore we cannot get more than six values of 𝜎, namely

𝜎, 1－𝜎, 𝜎/𝜎－1, 1/1－𝜎, 𝜎－1/𝜎, 1/𝜎,

and any symmetrical function of these will have the same value at any two equivalent places in the modular dissection (§ 33). Their sum is constant, but the sum of their squares may be put into the form

2(𝜎²－𝜎＋1)³/𝜎²(𝜎－1)²－3;

hence (𝜎²－𝜎＋1)³÷𝜎²(𝜎－1)² has the same value at equivalent places. F. Klein writes

𝖩＝4(𝜎²－𝜎＋1)³/27𝜎²(𝜎－1)²;

this is a transcendental function of 𝜔, which is a special case of a Fuchsian or automorphic function. It is an analytical function of 𝑞², and may be expanded in the form

𝖩＝1/1728{𝑞^－2＋744＋𝑐₁𝑞²＋𝑐₂𝑞⁴＋ . . . }

where 𝑐₁, 𝑐₂, &c., are rational integers.

69. Suppose, now, that 𝑎, 𝑏, 𝑐, 𝑑 are rational integers, such that dv(𝑎, 𝑏, 𝑐, 𝑑)＝1 and 𝑎𝑑－𝑏𝑐＝𝑛, a positive integer. Let (𝑎𝜔＋𝑏)/(𝑐𝜔＋𝑑)＝𝜔′; then the equation 𝖩(𝜔′)＝𝖩(𝜔) is satisfied if and only if 𝜔′∼ 𝜔, that is, if there are integers 𝛼, 𝛽, 𝛾, 𝛿 such that 𝛼𝛿－𝛽𝛾＝1, and

(𝑎𝜔＋𝑏)(𝛾𝜔＋𝛿)－(𝑐𝜔＋𝑑)(𝛼𝜔＋𝛽)＝0.

If we write 𝜓(𝑛)＝𝑛𝚷(1＋𝑝^－1), where the product extends to all prime factors (𝑝) of 𝑛, it is found that the values of 𝜔 fall into 𝜓(𝑛) equivalent sets, so that when 𝜔 is given there are not more than 𝜓(𝑛) different values of 𝖩(𝜔′). Putting 𝖩(𝜔′)＝𝖩′, 𝖩(𝜔)＝𝖩 we have a modular equation

𝑓₁(𝖩′, 𝖩)＝0

symmetrical in 𝖩, 𝖩′, with integral coefficients and of degree 𝜓(𝑛). Similarly when dv(𝑎, 𝑏, 𝑐, 𝑑)＝𝜏 we have an equation 𝑓_𝜏(𝖩′, 𝖩)＝0 of order 𝜓(𝑛/𝜏²); hence the complete modular equation for transformations of the 𝑛th order is

𝖥(𝖩′,𝖩)＝𝚷𝑓_𝜏(𝖩′, 𝖩)＝0,

the degree of which is 𝚽(𝑛), the sum of the divisors of 𝑛.

Now if in 𝖥(𝖩′, 𝖩) we put 𝖩′＝𝖩, the result is a polynomial in 𝖩 alone, which we may call 𝖦(𝖩). To every linear factor of 𝖦 corresponds a class of quadratic forms of determinant (𝜅²－4𝑛) where 𝜅²＜4𝑛 and 𝜅 is an integer or zero: conversely from every such form we can derive a linear factor (𝖩－𝛼) of 𝖦. Moreover, if with each form we associate its weight (§ 41) we find that with the notation of § 39 the degree of 𝖦 is precisely 𝚺𝖧(4𝑛－𝜅²)－𝜖_𝑛, where 𝜖_𝑛＝1 when 𝑛 is a square, and is zero in other cases. But this degree may be found in another way as follows. A complete representative set of transformations of order 𝑛 is given by 𝜔′＝(𝑎𝜔＋𝑏)/𝑑, with 𝑎𝑑＝𝑛, 0⩽𝑏＜𝑑; hence

$G(J)=\Pi \left\{J(\omega )-J\left({a\omega +b \over d}\right)\right\}$

and by substituting for 𝖩(𝜔) and $J\left({a\omega +b \over d}\right)$ their values in terms of 𝑞, we find that the lowest term in the factor expressed above is either 𝑞^－2/1728 or 𝑞^{－2𝑎/𝑑}/1728, or a constant, according as 𝑎＜𝑑, 𝑎＞𝑑 or 𝑎＝𝑑. Hence if 𝜈 is the order of 𝖦(𝖩), so that its expansion in 𝑞 begins with a term in 𝑞^－2𝜈 we must have

$\nu =\sum _{d>{\sqrt {n}}}(1\cdot d)+\sum _{d>{\sqrt {n}}}\left({a \over d}\cdot d\right)=\sum _{d>{\sqrt {n}}}d+\sum _{a>{\sqrt {n}}}a$

extending to all divisors of 𝑛 which exceed √𝑛. Comparing this with the other value, we have

$\sum _{\kappa }{\mbox{H}}(4n-\kappa ^{2})=2\sum _{d>{\sqrt {n}}}d+\epsilon _{n}=\Phi (n)+\Psi (n)$,

as stated in § 39.

70. Each of the singular moduli which are the roots of 𝖦(𝖩)＝0 corresponds to exactly one primitive class of definite quadratic forms, and conversely.

Corresponding to every given negative determinant －𝚫 there is an irreducible equation 𝜓(𝑗)＝0, where 𝑗＝1728𝖩, the coefficients of which are rational integers, and the degree of which is ℎ(－𝚫). The coefficient of the highest power of 𝑗 is unity, so that 𝑗 is an arithmetical integer, and its conjugate values belong one to each primitive class of determinant －𝚫. By adjoining the square roots of the prime factors of 𝚫 the function 𝜓(𝑗) may be resolved into the product of as many factors as there are genera of primitive classes, and the degree of each factor is equal to the number of classes in each genus. In particular, if {1, 1, 1/4(𝚫＋1)} is the only reduced form for the determinant －𝚫, the value of 𝑗 is a real negative rational cube. At the same time its approximate value is $\exp \left[-2\pi i\cdot {1+i{\sqrt {\Delta }} \over 2}\right]+744=744-e^{\pi {\sqrt {\Delta }}}$, so that, approximately, 𝑒^𝜋√𝚫＝𝑚³＋744 where 𝑚 is a rational integer. For instance 𝑒^𝜋√43＝884736743.9997775 . . . ＝ 960³＋744 very nearly, and for the class (1, 1, 11) the exact value of 𝑗 is －960³. Four and only four other similar determinants are known to exist, namely －11, －19, －67, －163, although thousands have been classified. According to Hermite the decimal part of 𝑒^𝜋√163 begins with twelve nines; in this case Weber has shown that the exact value of 𝑗 is －2¹⁸⋅3³⋅5³⋅23³⋅29³.

71. The function 𝑗(𝜔) is the most fundamental of a set of quantities called class-invariants. Let (𝑎, 𝑏, 𝑐) be the representative of any class of definite quadratic forms, and let 𝜔 be the root of 𝑎𝑥²＋𝑏𝑥＋𝑐＝0 which has a positive imaginary part; then 𝖥 (𝜔) is said to be a class-invariant for (𝑎, 𝑏, 𝑐) if $F\left({\alpha \omega +\beta \over \gamma \omega +\delta }\right)=F(\omega )$ for all real integers 𝛼, 𝛽, 𝛾, 𝛿 such that 𝛼𝛿－𝛽𝛾＝1. This is true for 𝑗(𝜔) whatever 𝜔 may be, and it is for this reason that 𝑗 is so fundamental. But, as will be seen from the above examples, the value of 𝑗 soon becomes so large that its calculation is impracticable. Moreover, there is the difficulty of constructing the modular equation 𝑓₁(𝖩, 𝖩′)＝0 (§ 69), which has only been done in the cases when 𝑛＝2, 3 (the latter by Smith in Proc. Lond. Math. Soc. ix. p. 242).

For moderate values of 𝚫 the difficulty can generally be removed by constructing algebraic functions of 𝑗. Suppose we have an irreducible equation

𝑥^𝑚＋𝑐₁𝑥^𝑚－1＋ . . . ＋𝑐_𝑚＝0,

the coefficients of which are rational functions of 𝑗(𝜔). If we apply any modular substitution 𝜔′＝𝖲(𝜔), this leaves the equation unaltered, and consequently only permutates the roots among themselves: thus if 𝑥₁(𝜔) is any definite root we shall have 𝑥₁(𝜔′)＝𝑥_𝑖(𝜔), where 𝑖 may or may not be equal to 1. The group of unitary substitutions which leave all the roots unaltered is a factor of the complete modular group. If we put 𝑦＝𝑥(𝑛𝜔), 𝑦 will satisfy an equation similar to that which defines 𝑥, with 𝑗′ written for 𝑗; hence, since 𝑗, 𝑗′ are connected by the equation 𝑓₁(𝑗, 𝑗′)＝0, there will be an equation 𝜓(𝑥, 𝑦)＝0 satisfied by 𝑥 and 𝑦. By suitably choosing 𝑥 we can in many cases find 𝜓(𝑥, 𝑦) without knowing 𝑓₁(𝑗, 𝑗′); and then the equation 𝜓(𝑥, 𝑥)＝0 defines a set of singular moduli, each one of which belongs to a certain value of 𝜔 and all the quantities derived from it by the substitutions which leave 𝑥(𝜔) unaltered.

As one of the simplest examples, let 𝑛＝2, 𝑥³－𝑗(𝜔)＝𝑦³－𝑗(𝜔′)＝0. Then the equation connecting 𝑥, 𝑦 in its complete form is of the ninth degree in each variable; but it can be proved that it has a rational factor, namely

𝑦³－𝑥²𝑦²＋495𝑥𝑦＋𝑥³－2⁴ . 3³ . 5³＝0,

and if in this we put 𝑥＝𝑦＝𝑢, the result is

𝑢⁴－2𝑢³－495𝑢²＋2⁴.3³.5³＝0,

the roots of which are 12, 20, － 15, － 15. It remains to find the values of 𝜔, to which they belong. Writing 𝛾₂(𝜔)＝∛𝑗, it is found that we may define 𝛾₂ in such a way that 𝛾₂(𝜔＋1)＝𝑒^{－2𝜋𝑖/3}𝛾₂(𝜔), 𝛾₂(－𝜔^－1)＝𝛾₂(𝜔), whence it is found that

$\gamma _{2}\left({\alpha \omega +\beta \over \gamma \omega +\delta }\right)=e^{-{2\pi i \over 3}(\gamma \delta +\gamma \alpha +\beta \delta -\beta \delta \gamma ^{2})}\gamma _{2}(\omega )$.

We shall therefore have 𝛾₂(2𝜔)＝𝛾₂(𝜔) for all values of [𝜔?] such that

$2\omega ={\alpha \omega +\beta \over \gamma \omega +\delta }$, 𝛼𝛿－𝛽𝛾＝1, 𝛾𝛿＋𝛾𝛼＋𝛽𝛿－𝛽𝛿𝛾²≡0 (mod 3).

Putting (𝛼, 𝛽, 𝛾, 𝛿)＝(0, －1, 1, 0) the conditions are satisfied, and 2𝜔＝𝑖√2. Now 𝑗(𝑖)＝172𝛿, so that 𝛾₂(𝑖)＝12; and since 𝑗(𝜔) is positive for a pure imaginary, 𝛾₂(𝑖√2)＝20. The remaining case is settled by putting

${\omega \over 2}={\alpha \omega +\beta \over \gamma \omega +\delta }$,

with 𝛼, 𝛽, 𝛾, 𝛿 satisfying the same conditions as before. One solution is (－1, 2, 1, 1) and hence 𝜔²＋3𝜔＋4＝0, so that $\gamma _{2}\left({-3+i{\sqrt {7}} \over 2}\right)=-15$.

Besides 𝛾₂, other irrational invariants which have been used with effect are 𝛾₃＝√(𝑗－172𝛿), the moduli 𝜅, 𝜅′, their square and fourth roots, the functions 𝑓, 𝑓₁, 𝑓₂ defined by

$f=2^{1 \over 6}(\kappa \kappa ')^{-{1 \over 12}}$, $f_{1}={\sqrt[{4}]{\kappa '}}\ .f,\ f_{2}={\sqrt[{4}]{\kappa }}.\,f$,

and the function 𝜂(𝑛𝜔)/𝜂(𝜔) where 𝜂(𝜔) is defined by

$\eta (\omega )=q^{1 \over 12}\sum _{-\infty }^{+\infty }(-1)^{s}q^{3s^{2}+s}={1 \over {\sqrt {3}}}\theta _{11}\left({2 \over 3},{\omega \over 3}\right)=q^{1 \over 12}\prod _{1}^{\infty }(1-q^{2s})$.

72. Another powerful method, developed by C. F. Klein and K. E. R. Fricke, proceeds by discussing the deficiency of 𝑓₁(𝑗, 𝑗′)＝0 considered as representing a curve. If this deficiency is zero, 𝑗 and 𝑗′ may be expressed as rational functions of the same parameter, and this replaces the modular equation in the most convenient manner. For instance, when 𝑛＝7, we may put

𝑗＝(𝜏²＋13𝜏＋49)(𝜏²＋5𝜏＋1)³/𝜏＝𝜙(𝜏), 𝑗′＝𝜙(𝜏′),

𝜏𝜏′＝49.

The corresponding singular moduli are found by solving 𝜙(𝜏)＝𝜙(𝜏′). For deficiency 1 we may find in a similar way two auxiliary functions 𝑥, 𝑦 connected by some simple equation 𝜓(𝑥, 𝑦)＝0 not exceeding the fourth degree, and such that 𝑗, 𝑗′ are each rational functions of 𝑥 and 𝑦.

Hurwitz has extended this field of research almost indefinitely, not only by generalising the formulae for class-number sums, such as that in § 69, but also by bringing the modular-function theory into connexion with that of algebraic correspondence and Abelian integrals. A comparatively simple example may help to indicate the nature of these researches. From the formulae given at the beginning of § 67, we can deduce, by actual multiplication of the corresponding series,

${1 \over \pi }\theta '_{11}\theta _{00}=\theta ^{2}{}_{00}\theta _{01}\theta _{10}=\sum _{-\infty }^{+\infty }\left({-1 \over \xi }\right)|\xi |q\xi ^{2}/_{4}\times \sum _{-\infty }^{+\infty }q\eta ^{2}\quad {\begin{cases}\xi =\pm 1,\ \pm 3,...;\\\eta =0,\ \ \pm 1,\ \ \pm 2,...\end{cases}}$

＝𝚺𝜒(𝑚)𝑞^𝑚/4    {𝑚＝1, 5, 9, . . .

where

$\chi (m)=\sum \left({-1 \over \xi }\right)|\xi |$

extended over all the representations 𝑚＝𝜉²＋4𝜂². In a similar way

1/𝜋𝜃′₁₁𝜃₁₀＝𝜃₀₀𝜃₁₀² 𝜃₀₁＝2𝚺(－1)^{1/4(𝑚－1)}𝜒(𝑚)𝑞^𝑚/2

1/𝜋𝜃′₁₁𝜃₁₀＝𝜃₀₀𝜃₁₀𝜃₀₁² ＝2𝚺(－1)^{1/4(𝑚－1)}𝜒(𝑚)𝑞^𝑚/4

If, now, we write

$j_{1}(\omega )=\sum {(-1)^{{1 \over 4}(m-1)}\chi (m) \over m}q^{m/2},\ \ \ j_{2}(\omega )=2\sum {(-1)^{{1 \over 4}(m-1)}\chi (m) \over m}q^{m/4}$,

$j_{3}(\omega )=2\sum {\chi (m) \over m}q^{m/4}$

we shall have

𝑑𝑗₁:𝑑𝑗₂:𝑑𝑗₃＝𝜃₁₀:𝜃₀₁:𝜃₀₀

where 𝜃₁₀, 𝜃₀₁, 𝜃₀₀ are connected by the relation (§ 67)

𝜃₁₀⁴＋𝜃₀₁⁴－𝜃₀₀⁴＝0

which represents, in homogeneous co-ordinates, a quartic curve of deficiency 3. For this curve, or any equivalent algebraic figure, 𝑗₁(𝜔), 𝑗₂(𝜔) and 𝑗₃(𝜔) supply an independent set of Abelian integrals of the first kind. If we put 𝑥＝√𝜅, 𝑦＝√𝜅′, it is found that

$\int {dx \over y^{3}}={1 \over 2}j_{3}(\omega ),\quad \int {dx \over y^{2}}={1 \over 2}j_{2}(\omega ),\quad \int {x\,dx \over y^{3}}={1 \over 2}j_{1}(\omega )$,

so that the integrals which the algebraic theory gives in connexion with 𝑥⁴＋𝑦⁴－1＝0 are directly identified with 𝑗₁(𝜔), 𝑗₂(𝜔), 𝑗₃(𝜔) provided that we put 𝑥＝√𝜅(𝜔).

Other functions occur in this theory analogous to 𝑗₁(𝜔), but such that in the 𝑞-series which are the expansions of them the coefficients and exponents depend on representations of numbers by quaternary quadratic forms.

73. In the Berliner Sitzungsberzchte for the period 1883–1890, L. Kronecker published a very important series of articles on elliptic functions, which contain many arithmetical results of extreme elegance; some of these Kronecker had announced without proof many years before. A few will be quoted here, without any attempt at demonstration; but in order to understand them, it will be necessary to bear in mind two definitions. The first relates to the Legendre-Jacobi symbol $\left({a \over b}\right)$. If 𝑎, 𝑏 have a common factor we put $\left({a \over b}\right)=0$; while if 𝑎 is odd and 𝑏＝2^ℎ𝑐, where 𝑐 is odd, we put $\left({a \over b}\right)=\left({2^{h} \over a}\right)\left({a \over c}\right)$. The other definition relates to the classification of discriminants of quadratic forms. If 𝖣 is any number that can be such a discriminant, we must have 𝖣≡0 or 1 (mod. 4), and in every case we can write 𝖣＝𝖣₀𝖰², where 𝖰² is a square factor of 𝖣, and 𝖣₀ satisfies one of the following conditions, in which 𝖯 denotes a product of different odd primes:—

		𝖣₀	＝	𝖯,	with	𝖯	≡	1 (mod 4)
		𝖣₀	＝	4𝖯,		𝖯	≡	－1 (mod 4)
		𝖣₀	＝	8𝖯,		𝖯	≡	±1 (mod 4)

Numbers such as 𝖣₀ are called fundamental discriminants; every discriminant is uniquely expressible as the product of a fundamental discriminant and a positive integral square.

Now let 𝖣₁, 𝖣₂ be any two discriminants, then 𝖣₁𝖣₂ is also a discriminant, and we may put 𝖣₁𝖣₂＝𝖣＝𝖣₀𝖰², where 𝖣₀ is fundamental: this being done, we shall have

$\tau \sum _{h=1}^{h=\infty }\sum _{k=1}^{k=\infty }\left({{\mbox{D}}_{1}{\mbox{Q}}^{2} \over h}\right)\left({{\mbox{D}}_{2}{\mbox{Q}}^{2} \over k}\right){\mbox{F}}(hk)$

$={1 \over 2}\sum _{a,b,c}\left[\left({{\mbox{D}}_{1} \over {\mbox{A}}}\right)+\left({{\mbox{D}}_{2} \over {\mbox{A}}}\right)\right]\sum _{m,n}\left({{\mbox{Q}}^{2} \over m}\right){\mbox{F}}(am^{2}+bmn+cn^{2})$

where we are to take ℎ, 𝑘＝1, 2, 3, . . .＋∞ ; 𝑚, 𝑛＝0, ±1, ±2, . . . ±∞ except that, if 𝖣＜0, the case 𝑚＝𝑛＝0 is excluded, and that, if 𝖣＞0, (2𝑎𝑚＋𝑏𝑛)𝖳⩾𝑛𝖴 where (𝖳, 𝖴) is the least positive solution of 𝖳²－𝖣𝖴²＝4. The sum $\sum _{a,b,c}$ applies to a system of representative primitive forms (𝑎, 𝑏, 𝑐) for the determinant 𝖣, chosen so that 𝑎 is prime to 𝖰, and 𝑏, 𝑐 are each divisible by all the prime factors of 𝖰. 𝖠 is any number prime to 2𝖣 and representable by (𝑎, 𝑏, 𝑐); and finally 𝜏＝2, 4, 6, 1 according as 𝖣＜－4, 𝖣＝－4, 𝖣＝－3 or 𝖣＞0. The function 𝖥 is quite arbitrary, subject only to the conditions that 𝖥(𝑥𝑦)＝𝖥(𝑥)𝖥(𝑦), and that the sums on both sides are convergent. By putting 𝖥(𝑥)＝𝑥^－1－𝜌, where 𝜌 is a real positive quantity, it can be deduced from the foregoing that, if 𝖣₂ is not a square, and if 𝖣₁ is different from 1,

$\tau {\mbox{H}}({\mbox{D}}_{1}{\mbox{Q}}^{2}){\mbox{H}}({\mbox{D}}_{2}{\mbox{Q}}^{2})=\lim _{\rho \rightarrow 0}\sum _{a,b,c}\left({{\mbox{D}}_{1} \over {\mbox{A}}}\right)\sum _{m,n}\left({{\mbox{Q}}^{2} \over m}\right)(am^{2}+bmn+cn^{2})^{-1-\rho }$

where the function 𝖧(𝑑) is defined as follows for any discriminant 𝑑:—

	𝑑＝－𝚫＜0	$\tau {\mbox{H}}(d)={2\pi \over {\sqrt {\Delta }}}h(-\Delta )$
	𝑑＞0	${\mbox{H}}(d)={h(d) \over 2{\sqrt {d}}}\log {{\mbox{T}}+{\mbox{U}}{\sqrt {d}} \over {\mbox{T}}-{\mbox{U}}{\sqrt {d}}}$

ℎ(𝑑) meaning the number of primitive forms for the determinant 𝑑. This is a generalisation of a theorem due to Dirichlet.

There is another formula which, in a certain sense, is the generalisation of Gauss’s sums (§ 62) in cyclotomy. Let 𝜓(𝑢, 𝑣) denote the function 𝜃₁₁(𝑢＋𝑣)÷𝜃₀₁(𝑢)𝜃₀₁(𝑣) and let 𝖣₁, 𝖣₂ be any two fundamental discriminants such that 𝖣₁𝖣₂ is also fundamental and negative: then

${\tau \theta '_{11} \over 2\pi |{\sqrt {{\mbox{D}}_{1}{\mbox{D}}_{2}}}|}\sum _{s_{1},\ s_{2}}\left({{\mbox{D}}_{1} \over s_{1}}\right)\left({{\mbox{D}}_{2} \over s_{2}}\right)\psi \left({2s_{1} \over |{\mbox{D}}_{1}|},{2s_{2} \over |{\mbox{D}}_{2}|}\right)$

$=\sum _{a,\ b,\ c}\left[\left({{\mbox{D}}_{1} \over {\mbox{A}}}\right)+\left({{\mbox{D}}_{2} \over {\mbox{A}}}\right)\right]\sum _{m,\ n}q^{{1 \over 2}(am^{2}+bmn+cn^{2})}$

where, on the left-hand side, we are to sum for 𝑠_𝑖＝1, 2, 3 . . . |𝖣_𝑖; and on the right we are to take a complete set of representative primitive forms (𝑎, 𝑏, 𝑐) for the determinant 𝖣₁𝖣₂, and give to 𝑚, 𝑛 all positive and negative integral values such that 𝑎𝑚²＋𝑏𝑚𝑛＋𝑐𝑛² is odd. The quantity 𝜏 is 2, if 𝖣₁𝖣₂＜－4, 𝜏＝4 if 𝖣₁𝖣₂＝－4, 𝜏＝6 if 𝖣₁𝖣₂＝－3. By putting 𝖣₂＝1, we obtain, after some easy transformations,

$\sum _{s=1}^{s=\Delta }\left({-\Delta \over s}\right){\mbox{sn}}{4s{\mbox{K}} \over \Delta }={4{\sqrt {\Delta }} \over \tau \theta _{10}{}^{2}}\sum \sum q^{{1 \over 2}(am^{2}+bmn+cn^{2})}$,

which holds for any fundamental discriminant －𝚫. For instance, taking 𝜔＝𝑖𝖪′/𝖪, and 𝚫＝3, we have 𝜃₁₀²＝2𝜅𝖪/𝜋, and 𝚺𝑞^{1/2(𝑚²＋𝑚𝑛＋𝑛²)}＝2𝜅𝖪√3/𝜋sn4𝖪/3; a verification is afforded by making 2𝖪 approach the value 𝜋, in which case 𝑞, 𝜅 vanish, while the limit of 𝑞^1/2/𝜅 is 1/4, whence the limiting value of sn4𝖪/3 is that of 6𝑞^1/2/𝜅√3, which ＝6/4√3＝√3/2, as it should be.

Several of Kronecker’s formulae connect the solution of the Pellian equation with elliptic modular functions: one example may be given here. Let 𝖣 be a positive discriminant of the form 8𝑛＋5, let (𝖳, 𝖴) be the least solution of 𝖳²－𝖣𝖴²＝1: then, if ℎ(𝖣) is the number of primitive classes for the determinant 𝖣,

(𝖳－𝖴√𝖣)^ℎ(𝖣)＝𝚷(2𝜅𝜅′)²

where the product on the right extends to a certain sixth part of those values of 2𝜅𝜅′ which are singular, and correspond to the field 𝛀(√－𝖣), or in other words are connected with the class invariant 𝑗(√－𝖣). For instance, if 𝖣＝5, the equation to find (𝜅𝜅′)² is

4⋅𝛿{(𝜅𝜅′)²－1}³＋(25＋13√5)³(𝜅𝜅′)⁴＝0

one root of which is given by (2𝜅𝜅′)²＝9－4√5＝𝖳－𝖴√5 which is right, because in this case ℎ(𝖣)＝1.

74. Frequency of Primes.—The distribution of primes in a finite interval (𝑎, 𝑎＋𝑏) is very irregular, if we change 𝑎 and keep 𝑏 constant. Thus if we put 𝑛!＝𝜇, the numbers 𝜇＋2, 𝜇＋3, . . . (𝜇＋𝑛－1) are all composite, so that we can form a run of consecutive composite numbers as extensive as we please; on the other hand, there is possibly no limit to the number of cases in which 𝑝 and 𝑝＋2 are both primes. Legendre was the first to find an approximate formula for 𝖥(𝑥), the number of primes not exceeding 𝑥. He found by induction

𝖥(𝑥)＝𝑥 ÷ (log_𝑒𝑥－1.08366)

which answers fairly well when 𝑥 lies between 100 and 1,000,000, but becomes more and more inaccurate as 𝑥 increases. Gauss found, by theoretical considerations (which, however, he does not explain), the approximate formula

${\mbox{F}}(x)={\mbox{L}}(x)=\int _{2}^{x}{dx \over \log x}$

(where, as in all that follows, log 𝑥 is taken to the base 𝑒). This value is ultimately too large, but when 𝑥 exceeds a million it is nearer the truth than the value given by Legendre’s formula.

By a singularly profound and original analysis, Riemann succeeded in finding a formula, of the same type as Gauss’s, but more exact for very large values of 𝑥. In its complete form it is very complicated; but, by omitting terms which ultimately vanish (for sufficiently large values of 𝑥) in comparison with those retained, the formula reduces to

${\mbox{F}}(x)={\mbox{A}}+\sum _{m}(-1)^{\mu }{1 \over m}{\mbox{L}}(x^{1/m})$

where the summation extends to all positive integral values of 𝑚 which have no square factor, and 𝜇 is the number of different prime factors of 𝑚, with the convention that when 𝑚＝1, (－1)^𝜇＝1. The symbol 𝖠 denotes a constant, namely

${\mbox{A}}=\sum {(-1)^{\mu } \over m}\times \left\{{1 \over 2}-\int _{2}^{\infty }{dx \over x(x^{2}-1)\log x}\right\}$

and 𝖫 is used in the sense given above.

P. L. Tchébichev obtained some remarkable results on the frequency of primes by an ingenious application of Stirling’s theorem. One of these is that there will certainly be (𝑘＋1) primes between 𝑎 and 𝑏, provided that

𝑎＜5𝑏/6 − 2√𝑏 − 16/25 𝖱 log 6 (log 𝑏)² − 5/24𝖱 (4𝑘＋25) − 25/6𝖱

where 𝖱 ＝ 1/2 log 2 ＋ 1/3 log 3 ＋ 1/5 log 5 − 1/30 log 30＝0.921292 . . . . From this may be inferred the truth of Bertrand’s conjecture that there is always at least one prime between 𝑎 and (2𝑎 − 2) if 2𝑎＞7. Tchébichev’s results were generalized and made more precise by Sylvester.

The actual calculation of the number of primes in a given interval may be effected by a formula constructed and used by D. F. E. Meissel. The following table gives the values of 𝖥(𝑛) for various values of 𝑛, according to Meissel’s determinations:—

𝑛	⁠	𝖥(𝑛)
20,000		2,262
100,000		9,592
500,000		41,538
1,000,000		78,498

Riemann’s analysis mainly depends upon the properties of the function

$\zeta (s)=\sum _{n}n^{-s}\qquad \qquad (n=1,\ 2,\ 3,\ .\ .\ .\ )$,

considered as a function of the complex variable 𝑠. The above definition is only valid when the real part of 𝑠 exceeds 1; but it can be generalized by writing

$2\sin \pi z\Gamma (z)\zeta (z)=i\int _{\infty }^{\infty }{\frac {(-x)^{z-1}dx}{e^{x}-1}}$

where the integral is taken from 𝑥＝＋∞ along the axis of real quantities to 𝑥＝𝜖, where 𝜖 is a very small positive quantity, then round a circle of radius 𝜖 and centre at the origin, and finally from 𝑥＝𝜖 to 𝑥＝＋∞ along the axis of real quantities. This function 𝜁(𝑧) is of great importance, and has been recently studied by von Mangoldt Landau and others.

Reference has already been made to the fact that if 𝑙, 𝑚 are coprimes the linear form 𝑙𝑥＋𝑚 includes an infinite number of primes. Now let (𝑎, 𝑏, 𝑐) be any primitive quadratic form with a total generic character 𝖢; and let 𝑙𝑥＋𝑚 be a primitive linear form chosen so that all its values have the character 𝖢. Then it has been proved by Weber and Meyer that (𝑎, 𝑏, 𝑐) is capable of representing an infinity of primes all of the linear form 𝑙𝑥＋𝑚.

75. Arithmetical Functions. -This term is applied to symbols such as 𝜙(𝑛), 𝚽(𝑛), &c., which are associated with 𝑛 by an intrinsic arithmetical definition. The function 𝚽(𝑛) was written ʃ𝑛 by Euler, who investigated its properties, and by proving the formula $\prod _{1}^{\infty }(1-q^{s})=\sum _{-\infty }^{+\infty }q^{{1 \over 2}3s^{2}+s}$ deduced the result that

$\int n=\int (n-1)+\int (n-2)-\int (n-5)-...=\sum (-1)^{s-1}\int \left(n-{3s^{2}\pm s \over 2}\right)$

where on the right hand we are to take all positive values of 𝑠 such that 𝑛－1/2(3𝑠²±𝑠) is not negative, and to interpret ʃ0 as 𝑛, if this term occurs. J. Liouville makes frequent use of this function in his papers, but denotes it by 𝜁(𝑛).

If the quantity 𝑥 is positive and not integral, the symbol E(𝑥) or [𝑥] is used to denote the integer (including zero) which is obtained by omitting the fractional part of 𝑥; thus E(√2) =1, E(0·7)=0, and so on. For some purposes it is convenient to extend the definition by putting E(-𝑥) = −E(𝑥), and agreeing that when 𝑥 is a positive integer, E(𝑥) =𝑥−1/2; it is then possible to find a Fourier sine-series representing 𝑥-E(𝑥) for all real values of 𝑥. The function E(𝑥) has many curious and important properties, which have been investigated by Gauss, Hermite, Hacks, Pringsheim, Stern and others. What is perhaps the simplest roof of the law of quadratic reciprocity depends upon the fact that if p, q are two odd primes, and we put p=2h+1, q=2k+1

$\sum _{r=1}^{r=h}{\mbox{E}}{\Big (}{\frac {rp}{q}}{\Big )}+\sum _{s=k}^{s=1}{\mbox{E}}{\Big (}{\frac {sp}{q}}{\Big )}=hk={\tfrac {1}{4}}(p-1)(q-1)$

the truth of which is obvious, if we rule a rectangle 5” Xq” into 'unit squares, and draw its diagonal. This formula is auss's, but the geometrical proof is due to Eisenstem. Another useful formula is

$\sum _{r=1}^{r=m-1}{\mbox{E}}{\Big (}x+{\frac {r}{m}}{\Big )}={\mbox{E}}(mx)-{\mbox{E}}(x)$, which is due to Hermite.

Various other arithmetical functions have been devised for particular purposes; two that deserve mention (both due to Kronecker) are 5;, ;, , which means 0 or 1 according as h, k are unequal or equal, and sgn 𝑥, which means 𝑥÷|𝑥|

76. Transcendental Numbers.-It has been proved by Cantor that the aggregate of all algebraic numbers is countable. Hence immediately follows the proposition (first proved by Liouville) that there are numbers, both real and complex, which cannot be defined b any combination of a finite number of equations with rationafiintegral coefficients. Such numbers are said to be transcendental. Hermite first completely proved the transcendent character of e; and Lindemann, by a similar method, proved the transcendence of -rr. Thus it is now finally established that the quadrature of the circle is impossible, not only by rule and compass, but even with the help of any number of algebraic curves of any order when the coefficients in their equations are rational (see Hermite, C.R. lxxvii., 1873, and Lindemann, Math. Ann. xx., 1882). Another number which is almost certainly transcendent is Euler's constant C. It may be convenient to give here the following numerical values:

π	＝3·14159	25535	89793	23846. . .
𝑒	＝2·71828	18284	59045	23536. . .
C	＝0·57721	56649	01532	8606065. . . (Gauss-Nicolai)
log10＝(π log10𝑒)		＝0·13493	41840. . .	(Weber),
the last of which is useful in calculating class-invariants.

77. Miscellaneous Investigations.—The foregoing articles (§§ 24-76) give an outline of what may be called the analytical theory of numbers, which is mainly the work of the 19th century, though many of the researches of Lagrange, Legendre and Gauss, as well as all those of Euler, fall within the 18th. But after all, the germ of this remarkable development is contained in what is only a part of the original Diophantine analysis, of which, beyond question, Fermat was the greatest master. The spirit of this method is still vigorous in Euler; but the appearance of Gauss’s Disquisitiones arithmeticae in 1801 transformed the whole subject, and gave it a new tendency which was strengthened by the discoveries of Cauchy, Jacobi, Eisenstein and Dirichlet. In recent times Edouard Lucas revived something of the old doctrine, and it can hardly be denied that the Diophantine method is the one that is really germane to the subject. Even the strange results obtained from elliptic and modular functions must somehow be capable of purely arithmetical proof without the use of infinite series. Besides this, the older arithmeticians have announced various theorems which have not been proved or disproved, and made a beginning of theories which are still in a more or less rudimentary stage. As examples of the latter may be mentioned the partition of numbers (see Numbers, Partition of, below), and the resolution of large numbers into their prime factors.

The general problem of partitions is to find all the integral solutions of a set of linear equations Σ𝑐_𝑖𝑥_𝑖＝𝑚_𝑖 with integral coefficients, and fewer equations than there are variables. The solutions may be further restricted by other conditions—for instance, that all the variables are to be positive. This theory was begun by Euler: Sylvester gave lectures on the subject, of which some portions have been preserved; and various results of great generality have been discovered by P. A. MacMahon. The author last named has also considered Diophantine inequalities, a simple problem in which is “to enumerate all the solutions of 7𝑥⪖13𝑦 in positive integers.”

The resolution of a given large number into its prime factors is still a problem of great difficulty, and tentative methods have to be applied. But a good deal has been done by Seelhoff, Lucas, Landry, A. J. C. Cunningham and Lawrence to shorten the calculation, especially when the number is given in, or can be reduced to, some particular form.

It is well known that Fermat was led to the erroneous conjecture (he did not affirm it) that 2^𝑚+1 is a prime whenever 𝑚 is a power of 2. The first case of failure is when 𝑚＝32; in fact 2³²+1 ≡ 0 (mod 641). Other known cases of failure are 𝑚＝2^𝑛, with 𝑛＝6, 12, 23, 26 respectively; at the same time, Eisenstein asserted that he had proved that the formula 2^𝑚+1 included an infinite number of primes. His proof is not extant; and no other has yet been supplied. Similar difficulties are encountered when We examine Mersenne’s numbers, which are those of the form 2^𝑝−1, with 𝑝 a prime; the known cases for which a Mersenne number is prime correspond to 𝑝＝2, 3, 5, 7, 13, 17, 19, 31, 61.

A perfect number is one which, like 6 or 28, is the sum of its aliquot parts. Euclid proved that 2^𝑝−1 (2^𝑝−1) is perfect when (2^𝑝−1) is a prime: and it has been shown that this formula includes all perfect numbers which are even. It is not known whether any odd perfect numbers exist or not.

Friendly numbers (numeri amicabiles) are pairs such as 220, 284, each of which is the sum of the aliquot parts of the other. No general rules for constructing them appear to be known, but several have been found, in a more or less methodical way.

78. In conclusion it may be remarked that the science of arithmetic (q.v.) has now reached a stage when all its definitions, processes and results are demonstrably independent of any theory of variable or measurable quantities such as those postulated in geometry and mathematical physics; even the notion of a limit may be dispensed with, although this idea, as well as that of a variable, is often convenient. For the application of arithmetic to geometry and analysis, see Function.

Authorities.—W. H. and G. E. Young, The Theory of Sets of Points (Cambridge, 1906; contains bibliography of theory of aggregates); P. Bachmann, Zahlentheorie (Leipzig, 1892; the most complete treatise extant); Dirichlet-Dedekind, Vorlesungen über Zahlentheorie (Braunschweig, 3rd and 4th ed., 1879, 18); K. Hensel, Theorie der algebraischen Zahlen (Leipzig, 1908); H. J. S. Smith, Report on the Theory of Numbers (Brit. Ass. Rep., 1859–1863, 1865, or Coll. Math. Papers, vol. i.); D. Hilbert, “Bericht über die Theorie der algebraischen Zahlkörper” (in Jahresber. d. deutschen Math.-Vereinig., vol. iv., Berlin, 1897); Klein-Fricke, Elliptische Modulfunctionen (Leipzig, 1890-1892); H. Weber, Elliptische Functionen u. algebraische Zahlen (Braunschweig, 1891). Extensive bibliographies will be found in the Royal Society’s Subject Index, vol. i. (Cambridge, 1908) and Encycl. d. math. Wissenschaften, vol. i. (Leipzig, 1898). (G. B. M.) 

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1. See also Numeral