238 HlSTOKY OF THE THEOKY OF NUMBERS. [CHAP. VIII degree e modulo p. Hence if p is a primitive root of n, f is irreducible modulo p, and therefore with respect to each of the infinitude of primes p+yn. Thus / is algebraically irreducible. Schonemann65 considered congruences modulo pm. If g(x) is not divisible by p} and/=rcn+ ... is irreducible modulo pm and if 4 (re) is not divisible by/ modulo p, thenfg=AB (mod pm) implies that B(x) is divisible by/ modulo pm. If /=/i, g=ffi (mod p) and the leading coefficients of the four functions are unity, while / and g have no common factor modulo p, then /0=figi(mod pm) implies f=fl9 g=Qi (mod pm). He proved the final theorem of Gauss.60 Next, (x-a)n+pF(x) is irreducible modulo p2 if and only if F(a)^0 (mod p} ; an example is Henceforth, let/fa) be irreducible modulo p and of degree n. If f(x)n+pF(x) is reducible modulo p2, then (p. 101) /(a) is a factor of F(z) modulo p. If /(a) = 0 and g(a) f^ 0 (mod p, a) , then tf= 1 (mod pm, a) , where e = pm~1(pn - 1) . If the roots of (?(«) are the (p™"""1)^ powers of the roots of /(re), then /2^ = ^ - fl v* - /W . . (z -/3pn~1) (mod pm, a) . has the leading coefficient unity, we can Divisible by F(x) modulo M. ess of the factorization of a function f(x) .cible factors modulo p, a prime. An irre- _____ j only when it divides one factor modulo p. ,.__ _ ...... _____ unctions divides their g. c. d. modulo p. Cauchy67 employed an indeterminate quantity or symbol i and defined f(i) to be not the value of the polynomial /(re) for x = i, but to be a+bi if a+bx is the remainder obtained by dividing /(x) by z2+l. In particular, if f(x) is x2+l itself, we have i2+l=0. Similarly, if co(o:) = 0 is an irreducible congruence modulo p, a prime, let i denote a symbolic root. Then <£(i)^({) = 0 implies either 0(i) = 0 or ^(i)sQ (mod p). At most n integral functions of i satisfy /(re, i) = 0 (mod p), if the degree of / in x is n<p. If our w(rr) divides rrn — 1, but not rcm-l, m<n, modulo p, where n is not a divisor of p — 1, call i a symbolic primitive root of of'ssl (mod p). Then xn-l=(x-l)(x-i') . . .(x-in~l.) If s is a primitive root of n and if n — I=gh, and p0= 1 (mod n), ;=0 equals a function of re with integral coefficients, while every factor of xn — 1 modulo p with integral coefficients equals such a product. "Jour, fur Math., 32, 1846, 93-105. B6Comptes Rendus Paris, 24, 1847, 1117; Oeuvrea, (1). 10, 308-12. "Comptes Rendus Paris, 24, 1847, 1120; Oeuvres, (1), 10, 312-23.