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.