234 History of the Theory of Numbers. [Chap, viii 
introduced in a way free from any logical objections. Avoiding their use, 
Gauss began his investigation by showing that two polynomials in xwith 
integral coefficients have a greatest common divisor modulo p, which can 
be found by Euclid's process. It is understood throughout that p is a 
prime (cf. Maser, p. 627). Hence if A and B are relatively prime poly- 
nomials modulo p, there exist two polynomials P and Q such that 
PA+QB^l {mod p). " 
Thus if A has no factor in common with B or C modulo p, we find by mul- 
tiplying the preceding congruence by C that A has no factor in common 
with the product BC modulo p. If a polynomial is divisible by A, B, C,. . ., 
no two of which have a common factor modulo p, it is divisible by their 
product. 
A polynomial is called prime modulo p if it has no factor of lower degree 
modulo p. Any polynomial is either prime or is expressible in a single 
way as a product of prime polynomials modulo p. The number of distinct 
polynomials x''+aa:"~^+ . . . modulo p is evidently p". Let (n) of these be 
prime functions. Then p'^-l^d{d), where d ranges over all the divisors of 
n (only a fragment of the proof is preserved). It is said to follow easily 
from this relation that, if n is a product of powers of the distinct primes 
a, 6, ... , then 
n(n)=p"-2p"/''+2:p"/''^- .... 
The rth powers of the roots of an equation P = with integral coefficients 
are the roots of an equation Pr = of the same degree with integral coeffi- 
cients. If r is a prime, P^=P (mod r). 
A prime function P of degree m, other than x itself, divides x' — l for 
some value of vKp"". If v is the least such integer, j^ is a divisor of p*" — 1. 
Hence P divides 
(1) x^"-^-l. 
The latter is congruent modulo p to the product of the prime functions, 
other than x, whose degrees are the various divisors of m. 
If P = x"'—Ax"'~^-{'Bx"'~^— ... is a prime function modulo p, the re- 
mainders by dividing the sum, the sum of the products by twos, etc., of 
^ ^p ~p' ^p"*~^ 
by P are congruent to A, B, etc., respectively. 
If V is not divisible by p and if m is the least positive integer for which 
^"•=1 (mod v), each prime function dividing x" — ! modulo p is a divisor of 
(1) and its degree is therefore a divisor of m. Let 6 be a divisor of m, and 
5', 8", ... the divisors <d oi 8; let ai be the g. c. d. of v and p^-1, fx' the 
g. c. d. of V and p*' — 1, . . . and set X' =iilii.' , \" =m/m", • • • • Then the num- 
ber of prime divisors modulo p of degree 6 of a:" — 1 is iV/5, if A^ is the num- 
ber of integers <y. which are divisible by no one of X', X", .... A method 
of finding all prime functions dividing j"— 1 is based on periods of powers 
of X with exponents < v and prime to v (pp. 620-2). 
