242 History of the Theory of Numbers. [Chap, viii 
proof is given for Dedekind's^^ 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 x" — 1 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 p" — 1 for }x<v. Let n be a product of powers of the 
distinct primes a, b,. . .. Then the product of all functions of degree v, 
irreducible modulo p, which belong to an exponent n which is a proper 
divisor of p" — 1, is congruent modulo p to 
n(a:"/"-l)-n(x"/"'"^-l)... 
and their number is therefore (f>{n)/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 /jl modulo p, which belong to 
the exponent 1= (p" — l)/<i, then if we replace x by x^, 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 Xfi which belong to the exponent \l, 
exception being made of the case when p is of the form 4ih — I, fi is odd, and 
X is divisible by 4. In this exceptional case, we may set p = 2H — l, i'^2, 
t odd; X = 2^s, j^2, s odd. Let k be the least of i, j. Then if we know 
the A^/2^~^ irreducible functions of odd degree ju modulo p which belong to 
the exponent I and if we replace x by x^, where X is of the form indicated, 
is prime to d and contains only primes dividing p" — 1, we obtain N/2^~^ 
functions of degree Xju each decomposable into 2^~'^ irreducible factors, 
thus giving A'' irreducible functions of degree \fx/2''~^ which belong to the 
exponent XL Apply these theorems to x — g% which belongs to the exponent 
(p — l)/d if ^ is a primitive root of p and if d is the g. c. d. of e and p — 1 ; we 
see that x^—g^ is irreducible unless the exceptional case arises, and is then 
a product of 2^"~^ irreducible functions. In that case, irreducible trinomials 
of degree X are found by decomposing x" —g% where i' = 2'~^X. 
If a is not divisible by p, a:'' — 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, 
(?(X, a:) = (mod p, F(a:)), 
we operate as if all multiples of F{x) vanish. There is here an indeter- 
minate 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(a:) = (mod p). From the theorems of the theory of func- 
tions modulis p, F{x), we may read off briefer theorems involving i (cf. 
Galois^2)^ 
