238 History of the Theory of Numbers. [Chap, viii 
degree e modulo p. Hence if p is a primitive root of n, / is irreducible 
modulo p, and therefore with respect to each of the infinitude of primes 
p+7«. Thus/ is algebraically irreducible. 
Schonemann^^ considered congruences modulo p"*. If g{x) is not divis- 
ible by p, and/=x''+ ... is irreducible modulo p"* and \i A{x) is not divisible 
by/ modulo p, then/g'=A5 (mod p'") implies that B{x) is divisible by/ 
modulo p"*. If /=/i, ^=6^1 (mod p) and the leading coefficients of the four 
functions are unity, while / and g have no common factor modulo p, then 
/f/^/i^iCinod p'") impHes /=/i, g=g\ (mod p"*). He proved the final 
theorem of Gauss. ^° Next, {x—aY-\-'pF{x) is irreducible modulo p^ if 
and only if F{a)^0 (mod p) ; an example is 
^ = {x-ir-'+pF{x), F(l) = l. 
Henceforth, let/(a;) be irreducible modulo p and of degree n. If f{xY-\-'pF{x) 
is reducible modulo p", then (p. 101) /(x) is a factor of F(a;) modulo p. If 
/(a) = and g(a) ^0 (mod p, a), then g'^ 1 (mod p'", a), where e = p'"~Hp''- 1). 
If the roots of G{z) are the (p'"~^)th powers of the roots of /(x), then 
G{z)^{z-^){z-n...{z-^'''-') (mod p-, a). 
If M is any integer and if F{x) has the leading coefficient unity, we can 
find z and w such that {x^—iy is divisible by F(x) modulo M. 
A. Cauchy^® noted the uniqueness of the factorization of a function f{x) 
with integral coefficients into irreducible factors modulo p, a prime. An irre- 
ducible function divides a product only when it divides one factor modulo p. 
A common divisor of two functions divides their g. c. d. modulo p. 
Cauchy^^ employed an indeterminate quantity or symbol i and defined 
f{i) to be not the value of the polynomial f{x) for x = i, but to be a-\-hi if 
a-\-hx is the remainder obtained by dividing /(a:) by x^+1. In particular, 
if /(x) is x^+1 itself, we have i^+1 =0. 
Similarly, if w(x) = is an irreducible congruence modulo p, a prime, 
let i denote a sjmiboHc root. Then 0(i);/'(t) = O implies either </)(i) = or 
yp{i) = (mod p). At most n integral functions of i satisfy /(x, i) = (mod p), 
if the degree of / in a: is nKp. If our co(x) divides x" — 1, but not x"* — 1, 
m<n, modulo p, where n is not a divisor of p — 1, call i a symbolic primi- 
tive root of x''=l (mod p). Then rc"-l=(.T-l)(x-i) . . . (x-i"-^) If 
s is a primitive root of n and if n — l=gh, and p''= 1 (mod n), 
equals a function of x with integral coefficients, while every factor of x" — 1 
modulo p with integral coefficients equals such a product. 
«Jour. fur Math., 32, 1846, 93-105. 
"Comptes Rendus Paris, 24, 1847, 1117; Oeuvres, (1). 10, 308-12. 
"Comptes Rendus Paris, 24, 1847, 1120; Oeuvres, (1), 10, 312-23. 
