250 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. VIII then f(x) is congruent modulo pa to a product of FI(X),. . ., Ft(x), each irreducible modulo pa, such that Fi(x)=fi(xY* (mod p), where fWs-Uffa)** (mod p), and /»(o;) is irreducible modulo p. There is an example of an irreducible cyclotomic function reducible with respect to every prime power modulus. P. Bachmann104 gave an exposition of the general theory. G. Arnoux105 exhibited in the form of tables the work of finding a primitive root of the GF[73] and of the GF[&], and tabulated the reducible and irreducible congruences of degrees 1, 2, 3, modulo 5. S. Epsteen106 proved the result of Guldberg,97 and developed the theory of residues of linear differential forms parallel to the theory of finite fields, as presented by Dickson.102 L. E. Dickson107 noted that the last mentioned subjects are identical abstractly. Let the irreducible form Ay be To the element (4) we make correspond the element Sc^ of the Galois field of order pn, where z is a root of the irreducible congruence dnzn+... +813+5<psO (mod p). Since product relations are preserved by this correspondence, the pn residues (4) define a field abstractly identical with our Galois field. Dickson107" proved that xmz=x (mod w = pn) has p and only p roots if p is a prime and hence does not define the Galois field of order m as occasionally stated. A. Guldberg1076 employed the notation of finite differences and wrote i-O where 6ys=yx^.it 62yx = yx+2^ . ., symbolically. To these linear forms with integral coefficients taken modulo p, a prime, we may apply Euclid's g. c. d. process and prove that factorization is unique. Next, let bm be not divisible by p, so that Gyx is of order m. With respect to the two moduli p, Gyx) a complete set of pm residues of linear forms is am_12/x+m_1+ . . . +aQyx (a,- = 0, 1,..., p — 1). Amongst these occur <t>(Gyx) =pw(l — l/pmi) . . .(1 — 1/?A) forms Fyx prime to Gyx iimi,...,mq are the orders of the irreducible factors of Gyx modulo p, and In particular, if Gyx is irreducible and of order m, 1MNiedere ZahJentheorie, 1, 1902, 363-399. 108As80C. frang. av. sc., 31, 1902, II, 202-227. 106Bull. Amer. Math. Soc., 10, 1903-4, 23-30. wjbid., pp. 30-1. 107flAmer. Math. Monthly. 11. 1904. 39-40.