Page:EB1911 - Volume 19.djvu/887

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