242 HISTORY OF THE THBOBT OP NTTMBEBS. [CHAP, vm proof is given for Dedekind's71 final theorem on the product of all irreducible functions of degree m modulo p. A function F(x) of degree v, irreducible modulo p, is said to belong to the exponent n if n is the least positive integer such that xn—l is divisible by F(x) modulo p. Then n is a divisor of p"— 1, and a proper divisor of it, since it does not divide pM— 1 f or M< v. Let n be a product of powers of the distinct primes a, 6, .... Then the product of all functions of degree v, irreducible modulo p, which belong to an exponent n which is a proper divisor of pv— 1, is congruent modulo p to and their number is therefore <l>(ri)/v. By a skillful analysis, Serret obtained theorems of practical importance for the determination of irreducible congruences of given degrees. If we know the N irreducible functions of degree /z modulo p, which belong to the exponent 1= (p"—l)/d, then if we replace x by xx, where X is prime to d and has no prime factor different from those which divide p"— 1, we obtain the N irreducible functions of degree X/x which belong to the exponent \l, exception being made of the case when p is of the form 4h — 1, ^ is odd, and X is divisible by 4. In this exceptional case, we may set p = 2it— 1, i^.2, t odd; X=2ys, j'^2, s odd. Let k be the least of i, j. Then if we know the N/2k~l irreducible functions of odd degree p modulo p which belong to the exponent I and if we replace x by xx, where X is of the form indicated, is prune to d and contains only primes dividing pM— 1, we obtain N/2k~l functions of degree X/z each decomposable into 2*""1 irreducible factors, thus giving N irreducible functions of degree XAi/2^1 which belong to the exponent XL Apply these theorems to x— ge, which belongs to the exponent (p — l)/d if g is a primitive root of p and if d is the g. c. d. of e and p — 1 ; we see that sx— ge is irreducible unless the exceptional case arises, and is then a product of 2k~l irreducible functions. In that case, irreducible trinomials of degree X are found by decomposing xv— ge, where v = 2i~1X. If a is not divisible by pf xp— x — a is irreducible modulo p. There is a development of Dedekind's theory of functions modulis p and F(x), where F(x) is irreducible modulo p. Finally, that theory is considered from the point of view of Galois. Just as in the theory of congruences of integers modulo p we treat all multiples of p as if they were zero, so in congruences in the unknown X, we operate as if all multiples of F(x) vanish. There is here an indeterminate x which we can make use of to cause the multiples of F(x) to vanish if we agree that this indeterminate x is an imaginary root i of the irreducible congruence F(x) = Q (mod p). From the theorems of the theory of functions modulis p} F(x), we may read off briefer theorems involving i (cf. Galois62).