1911 Encyclopædia Britannica/Number/Algebraic Numbers

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 $\eta ^{2}+\eta +6=0$, the roots of which are expressible as rational integral functions of 23rd roots of unity, and let $\eta$ be either of the roots. If we define $a\eta +b$ to be an integer, when $a,b$ are natural numbers, the product of any number of such integers is uniquely expressible in the form $l\eta +m$. Conversely every integer can be expressed as the product of a finite number of indecomposable integers $a\eta +b$, that is, integers which cannot be further resolved into factors of the same type. But this resolution is not necessarily unique: for instance $6=2.3=-\eta (\eta +1)$, where $2,3,\eta ,\eta +1$ are all indecomposable and essentially distinct. To see the way in which Kummer surmounted the difficulty consider the congruence

$u^{2}+u+6\equiv 0{\pmod {p}}$

where $p$ is any prime, except $23$. If $-23\mathrm {R} p$ this has two distinct roots $u_{1},u_{2}$; and we say that $a\eta +b$ is divisible by the ideal prime factor of $p$ corresponding to $u_{1}$, if $au_{1}+b\equiv 0{\pmod {p}}$. For instance, if $p=2$ we may put $u_{1}=0,u_{2}=1$ and there will be two ideal factors of $2$, say $p_{1}$ and $p_{2}$ such that $au_{1}+b\equiv 0{\pmod {p_{1}}}$ if $b\equiv 0{\pmod {2}}$ and $au_{1}+b\equiv 0{\pmod {p_{2}}}$ if $a+b\equiv 0{\pmod {2}}$. If both these congruences are satisfied, $a\equiv b\equiv 0{\pmod {2}}$ and $a\eta +b$ is divisible by 2 in the ordinary sense. Moreover $(a\eta +b)(c\eta +d)=(bc+ad-ac)\eta +(bd-6ac)$ and if this product is divisible by $p_{1}$, $bd\equiv 0{\pmod {2}}$, whence either $a\eta +b$ or $c\eta +d$ is divisible by $p_{1}$; while if the product is divisible by $p_{2}$ we have $bc+ad+bd-7ac=0{\pmod {2}}$ which is equivalent to $(a+b)(c+d)\equiv 0{\pmod {2}}$, so that again either $a\eta +b$ or $c\eta +d$ is divisible by $p_{2}$. Hence we may properly speak of $p_{1}$ and $p_{2}$ as prime divisors. Similarly the congruence $u^{2}+u+6\equiv 0{\pmod {3}}$ defines two ideal prime factors of 3, and $a\eta +b$ is divisible by one or the other of these according as $b\equiv 0{\pmod {3}}$ or $2a+b\equiv 0{\pmod {3}}$; we will call these prime factors $p_{3},p_{4}$. With this notation we have (neglecting unit factors)

$2=p_{1}p_{2},3=p_{3}p_{4},\eta =p_{1}p_{3},1+\eta =p_{2}p_{4}$.

Real primes of which $-23$ is a non-quadratic residue are also primes in the field $(\eta )$; and the prime factors of any number $a\eta +b$, as well as the degree of their multiplicity, may be found by factorizing $(6a^{2}-ab+b^{2})$, the norm of $a\eta +b$. Finally every integer divisible by $p_{2}$ is expressible in the form $\pm 2m\pm (1+\eta )n$ where $m,n$ are natural numbers (or zero) ; it is convenient to denote this fact by writing $p_{2}=[2,1+\eta ]$, and calling the aggregate $2m+(1+\eta )n$ a compound modulus with the base $2,1+\eta$. This generalized idea of a modulus is very important and far-reaching; an aggregate is a modulus when, if $\alpha ,\beta$ are any two of its elements, $\alpha +\beta$ and $\alpha -\beta$ also belong to it. For arithmetical purposes those moduli are most useful which can be put into the form $[\alpha _{1},\alpha _{2},\dots \alpha _{n}]$ which means the aggregate of all the quantities $x_{1}\alpha _{1}+x_{2}\alpha _{2}+\dots +x_{n}\alpha _{n}$ obtained by assigning to $(x_{1},x_{2},\dots x_{n})$, independently, the values $0,\pm 1,\pm 2\!\!\And \!\!\!\!\mathrm {c} .$ Compound moduli may be multiplied together, or raised to powers, by rules which will be plain from the following example. We have

${\begin{aligned}{p_{2}}^{2}&=[4,2(1+\eta ),(1+\eta )^{2}]=[4,2+2\eta ,-5+\eta ]=[4,12,-5+\eta ]\\&=[4,-5+\eta ]=[4,3+\eta ]\\\end{aligned}}$

hence

${\begin{aligned}{p_{2}}^{3}&={p_{2}}^{2}.p_{2}=[4,3+\eta ]\times [2,1+\eta ]=[8,4+4\eta ,6+2\eta ,3+4\eta +\eta ^{2}]\\&=[8,4+4\eta ,6+2\eta ,-3+3\eta ]=(\eta -1)[\eta +2,\eta -6,3]=(\eta -1)[1,\eta ].\\\end{aligned}}$

Hence every integer divisible by ${p_{2}}^{3}$ is divisible by the actual integer $(\eta -1)$ and conversely; so that in a certain sense we may regard $p_{2}$ as a cube root. Similarly the cube of any other ideal prime is of the form $(a\eta +b)[1,\eta ]$. 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 $a_{1},a_{2}.\dots a_{n}$ are ordinary integers (i.e. elements of $\mathrm {N}$, §7), and $\theta$ satisfies an equation of the form

$\theta ^{n}+a_{1}\theta ^{n-1}+a_{2}\theta ^{n-2}+\dots +a_{n-1}\theta +a_{n}=0$,

$\theta$ is said to be an algebraic integer. We may suppose this equation irreducible; $\theta$ is then said to be of the $n{\mbox{th}}$ order. The $n$ roots $\theta ,\theta ',\theta '',\dots \theta ^{(n-1)}$ are all different, and are said to be conjugate. If the equation began with $a_{0}\theta ^{n}$ instead of $\theta ^{n}$, $\theta$ would still be an algebraic number; every algebraic number can be put into the form $\theta /m$, where $m$ is a natural number and $\theta$ an algebraic integer.

Associated with $\theta$ we have a field (or corpus) $\Omega =\mathrm {R} (\theta )$ consisting of all rational functions of $\theta$ with real rational coefficients; and in like manner we have the conjugate fields $\Omega '=\mathrm {R} (\theta ')$, &c. The aggregate of integers contained in $\Omega$ is denoted by ${\mathfrak {o}}$.

Every element of $\Omega$ can be put into the form

$\omega =c_{0}+c_{1}\theta +\dots +c_{n-1}\theta ^{n-1}$

where $c_{0},c_{1},\dots c_{n-1}$ are real and rational. If these coefficients are all integral, $\omega$ is an integer; but the converse is not necessarily true. It is possible, however, to find a set of integers $\omega _{1},\omega _{2},\dots \omega _{n}$ belonging to $\Omega$, such that every integer in $\Omega$ can be uniquely expressed in the form

$h_{1}\omega _{1}+h_{2}\omega _{2}+\dots +h_{n}\omega _{n}$

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 $\mathrm {N} (m)=m^{n}$.