CHAP, viii) HIGHER CONGRUENCES, GALOIS IMAGINARIES. 249 If the product is =*Cy (mod p), Ay and By are called divisors modulo p of Cy. Let AT/ be of order n and irreducible modulo p. Then Ay is congruent modulis p, Ay to one and but one of the pn forms n— 1 $*! (4) Zc^ (Ci=0,l,...)P-l). If u is any one of these forms (4) and if e=pn— 1, Guldberg stated the analogue of Fermat's theorem ^=u (mod p, Ay), but incorrectly gave the right member to be unity [cf . Epsteen,106, Dickson107]. L. Stickelberger98 considered F(x) =xn+aiXn~1+ . . . with integral coefficients, such that the product D of the squares of the differences of the roots is not zero. Let p be any prime not dividing D. Let v be the number of factors of F(x) which are irreducible modulo p. He proved by the use of prune ideals that (!)=<-»-. where the symbol in the left member is that of Legendre [see quadratic residues], L. E. Dickson" proved the existence of the Galois field GF\pr] of order pn by induction from r — n to r=qn, by showing that is a product of factors of degree q belonging to and irreducible in the GF\pn]. Any such factor defines the GF[pnq]. L. Kronecker100 treated congruences modulis p, P(x) from the standpoint of modular systems. F. S. Carey101 gave for each prime p< 100 a table of the residues of the first p+1 powers of a primitive root a+bj of zv*~l=l (mod -p) where j2=j> (mod p), v being an integral quadratic non-residue of p. The higher powers are readily derived. While only the single modulus p is exhibited, it is really a question of a double modulus p and or2 — v. Methods of "solving" zPn-i~l are Discussed. in particular, for n = 3, there is given a primitive root for each prime p< 100. L. E. Dickson102 gave a systematic introductory exposition of the theory, with generalizations and extensions. M. Bauer103 proved that, if f(x) =0 is an irreducible equation with integral coefficients and leading coefficient unity, w a root, D its discriminant, d = D/k2 that of the domain defined by w, p & prime not dividing k, x>I, 98Verhand. I. Internat. Math. Kongress, 1897, 186. "Bull. Amer. Math. Soc., 6, 1900, 203-4. 100Vorlesungen iiber Zahlentheorie, I, 1901, 212-225 (expanded by Hensel, p. 506). "'Proc. London Math. Soc., 33, 1900-1, 294-310. "'Linear groups with an exposition of the Galois field theory, Leipzig, 1901, pp. 1-71. 10JMath. Naturw. Berichte aus Ungarn, 20, 1902, 39-42; Math. 6s PhyB. Lapok, 10, 1902, 28-33.