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
, the roots of which are expressible as rational integral functions of 23rd roots of unity, and let
be either of the roots. If we define
to be an integer, when
are natural numbers, the product of any number of such integers is uniquely expressible in the form
. Conversely every integer can be expressed as the product of a finite number of indecomposable integers
, that is, integers which cannot be further resolved into factors of the same type. But this resolution is not necessarily unique: for instance
, where
are all indecomposable and essentially distinct. To see the way in which Kummer surmounted the difficulty consider the congruence
![{\displaystyle u^{2}+u+6\equiv 0{\pmod {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08b0c9285dde35d0b11e33e2c3820f50f0c0b875)
where
is any prime, except
. If
this has two distinct roots
; and we say that
is divisible by the ideal prime factor of
corresponding to
, if
. For instance, if
we may put
and there will be two ideal factors of
, say
and
such that
if
and
if
. If both these congruences are satisfied,
and
is divisible by 2 in the ordinary sense. Moreover
and if this product is divisible by
,
, whence either
or
is divisible by
; while if the product is divisible by
we have
which is equivalent to
, so that again either
or
is divisible by
. Hence we may properly speak of
and
as prime divisors. Similarly the congruence
defines two ideal prime factors of 3, and
is divisible by one or the other of these according as
or
; we will call these prime factors
. With this notation we have (neglecting unit factors)
.
Real primes of which
is a non-quadratic residue are also primes in the field
; and the prime factors of any number
, as well as the degree of their multiplicity, may be found by factorizing
, the norm of
. Finally every integer divisible by
is expressible in the form
where
are natural numbers (or zero) ; it is convenient to denote this fact by writing
, and calling the aggregate
a compound modulus with the base
. This generalized idea of a modulus is very important and far-reaching; an aggregate is a modulus when, if
are any two of its elements,
and
also belong to it. For arithmetical purposes those moduli are most useful which can be put into the form
which means the aggregate of all the quantities
obtained by assigning to
, independently, the values
Compound moduli may be multiplied together, or raised to powers, by rules which will be plain from the following example. We have
![{\displaystyle {\begin{aligned}{p_{2}}^{2}&=[4,2(1+\eta ),(1+\eta )^{2}]=[4,2+2\eta ,-5+\eta ]=[4,12,-5+\eta ]\\&=[4,-5+\eta ]=[4,3+\eta ]\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2e7a13782db18550ca93fce9e4ede5bb7560a62)
hence
![{\displaystyle {\begin{aligned}{p_{2}}^{3}&={p_{2}}^{2}.p_{2}=[4,3+\eta ]\times [2,1+\eta ]=[8,4+4\eta ,6+2\eta ,3+4\eta +\eta ^{2}]\\&=[8,4+4\eta ,6+2\eta ,-3+3\eta ]=(\eta -1)[\eta +2,\eta -6,3]=(\eta -1)[1,\eta ].\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc7bfd7e1f38912240726e2e16130a47a9f9b69)
Hence every integer divisible by
is divisible by the actual integer
and conversely; so that in a certain sense we may regard
as a cube root. Similarly the cube of any other ideal prime is of the form
. 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
there are three primitive classes, represented by
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
are ordinary integers (i.e. elements of
, §7), and
satisfies an equation of the form
,
is said to be an algebraic integer. We may suppose this equation irreducible;
is then said to be of the
order.
The
roots
are all different, and are said to be conjugate.
If the equation began with
instead of
,
would still be an algebraic number; every algebraic number can be put into the form
, where
is a natural number and
an algebraic integer.
Associated with
we have a field (or corpus)
consisting of all rational functions of
with real rational coefficients; and in like manner we have the conjugate fields
, &c. The aggregate of integers contained in
is denoted by
.
Every element of
can be put into the form
![{\displaystyle \omega =c_{0}+c_{1}\theta +\dots +c_{n-1}\theta ^{n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a19e9ec4ac6d0b3b7e7e88993f05e935f84e5603)
where
are real and rational. If these coefficients are all integral,
is an integer; but the converse is not necessarily true. It is possible, however, to find a set of integers
belonging to
, such that every integer in
can be uniquely expressed in the form
![{\displaystyle h_{1}\omega _{1}+h_{2}\omega _{2}+\dots +h_{n}\omega _{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7243173dc394d53c9a3890dab0ee8da29750d720)
where
are elements of
which may be called the co-ordinates of
with respect to the base
. Thus
is a modulus (§ 44), and we may write
. Having found one base, we can construct any number of equivalent bases by means of equations such as
, where the rational integral coefficients
are such that the determinant
.
If we write
![{\displaystyle \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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bdd4cda84098aa75b506e0102d8787c152020f)
is a rational integer called the discriminant of the field. Its value is the same whatever base is chosen.
If
is any integer in
, the product of
and its conjugates is a rational integer called the norm of
, and written
. By considering the equation satisfied by
we see that
where
is an integer in
. It follows from the definition that if
are any two integers in
, then
; and that for an ordinary real integer
, we have
.