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=f=bi) =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, ri are any two complex integers and N(m) >N(n), we can always find integers q, r such that m=gn+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 (I +i)', (I +i) or is prime to (1+i). Among four associated odd integers there is one and only one whichré I (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 bEo (mod 2) a+b-120 (mod 4). Every complex integer can be uniquely expressed in the form i"'(I+i)"a°bBc1' . ., where 0 5m<4, and a, b, c, . are primary primes.

With respect to a complex modulus m, all complex integers may be distributed into N (m) congruous 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, I, 2, .. .{(a'+b')h-Il; y=o, I, 2, (h-I). Thus when b=o we may take x=o, 1, 2, ...(h-I); y=o, I, 2, . . (h-1), giving the h' residues of the real number h; while if a-|-bi is prime, r, 2, 3, . . .(a'+b2-I) form a complete set of residues.

The number of residues of m that are prime to m is given by ¢<m) =N(m>n (1 -NQB)

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¢<"'>E I (mod m), xN

°1i=- I (mod p) according as m is composite or prime. There are ¢lN(p) -I) primitive roots of the prime p; a primitive root in the real theory for a real prime 4n+I is also a primitive root in the new theory for each prime factor of (4n+I), 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 zand 'by the congruences p{{N(q)-tl; 2, p1lN(q)'llE= '(m0d Q), it being understood that the symbols stand for absolutely least P residues. It follows that 2 = I or - I according as p is a quadratic residue of q or not; and that '=I only if p is a bi quadratic residue of q. If p, q are primary primes, we have two laws of reciprocity, expressed by the equations p/q.-al q/p=N<»»~1=-=f-To these must be added the supplementary formulae 2=<-l)i{N<»>-Il, 2=(-l)a1<»+b>=-=l, 4=ie, <a-1) '=i}{a+b-<l+b>2}, a+bi being a primary odd prime. In words, the law of bi quadratic reciprocity for two primary odd primes mayfbe expressed by saying that the biquadrate characters of each prime with respect to the other are identical, unless p=qE3-|-2i (mod 4), in which case they are opposite. The law of biqkpadratic reciprocity was discovered by Gauss, who does not seem, owever, to have obtained a complete proof of it. The first published proof is that of Eiseustein, 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.—Thefirst 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 fl' +11 +6 =0, the roots of which are expressible as rational integral functions of 23rd roots of unity, and let 17 be either of the roots. If we define an+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 lf;-I-m. Conversely evei}y integer can be expressed as the product of a finite number o in decomposable 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

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 ul, if aul+bE0 (mod p). For instance, if p=2 we may put ul=o, ul= I and there will be two ideal factors of 2, say pl and pg such that an+b≡0 (mod pl) if bEo (mod 2) and a1;+bEo (mod pl) if a+bEo (mod 2). If both these congruences are satisfied, a==° bio (mod 2) and af;+b is divisible by 2 in the ordinary sense. Moreover (a1;+b)(c17+d)=(bc+ad-ac)»q+(bd-6ac) and if this product is divisible by pl, bdEo (mod 2), whence either an+b or cv;+d is divisible by pl; while if the product is divisible by pg we have bc+ad+bd-7ac==o (mod 2) which is equivalent to (a-HJ) (c +d)§ o (mod 2), so that again either a17~l-b or 617+d is divisible by pl. Hence we may properly speak of pl and pl as [prime divisors. Similarly the congruence u'+u+620 (mod 3) defines two ideal prime factors of 3, and ar;+b is divisible by one or the other of these according as bEo (mod 3) or 2a+bEo (mod 3); we will call these prime factors pl, p¢. With this notation we have (neglecting unit factors)

2 =P1Pzl 3 '-=PsP4l 17 =P1P3l I +7 =172P4

Real primes of which -23 is a non-quadratic residue are also primes in the field (1l); and the prime factors of any number an-|-b, as well as the degree of their multiplicity, may be found by factorizing (6a'-ab+bz), the norm of (an+b). Finally every integer divisible by pl is expressible in the form =b 2m i (I +17)n where m, n are natural numbers (or zero); it is convenient to denote this fact by writing pl=[2, 1+-n], and calling the aggregate 2711-|~(I-i-'l])?l a compound modulus with the base 2, I +1l. This generalized idea of a modulus is very important and far-reaching; an aggregate is a modulus when, if a, B are any two of its elements, a+H and a-B also belong to it. For arithmetical purposes those moduli are most useful which can be put into the form [al, a2, ...a., ,] which means the aggregate of all the quantities xlal+x2al+...+x, .a, . obtained by assigning to (xl, xl, ...x, ,), independently, the values ol=|=I, =|=2, &c. Compound moduli may be multiplied together, or raised to powers, by rules which will be plain from the following example. We have

pl==[4, 2<1+n). (1 'l'1l)2]=1412+2171-5+1ll=[4l 12, -5+n] =l4l-5+1:] =[4l 3+11]

hence

P2'=P2'-P2=l4» 3-1-11] ><l2l I +nl=[S»4+4nl 6-l~211»3+411+112] =l8, 4+41l. 6+2n, -3+3111=(11-I)[11+2,11-6,3]=(11-I)[I.11l-Hence every integer divisible by pl” is divisible by the actual integer (11-I) and conversely; so that in a certain sense we may regard pz as a cube root. Similarly the cube of any other ideal prime is of the form (a1l+b)[I, 1;]. 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, 5) 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, whic h is as follows: if al, al, ...a, . are ordinary integers (i.e. elements of N, § 7), and 0 satisfies an equation of the form

6"'i'a10" l'l'a20” 2'i" - - - 'l'an»-16'l'an:ol

${\displaystyle \theta }$ is said to be an algebraic integer. We may suppose this equation irreducible; ${\displaystyle \theta }$ is then said to be of the nth order. The n roots 0, 0', 0”, . .0<""1) are all different, and are said to be conjugate. If the equation began with (1/00" instead of 0", 6 would still be an algebraic number; every algebraic number can be put into the form 6/m, where m is a natural number and 0 an algebraic integer. Associated with 6 we have afield (or corpus) S2 = R(0) consisting of all rational functions of 0 with real rational coefficients; and in like manner we have the conjugate fields S2' = R(»9'), &c. The aggregate of integers contained in S2 is denoted by o.

Every element of Ω can be put into the form

where c₀, cl, ...c, , l are real and rational. If these coefficients are all integral, w 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