CHAP, viii] HIGHER CONGEUENCES, GALOIS IMAGINARIES. 247
E. H. Moore89 stated that every finite field (Korper) is, apart from notations, a Galois field composed of the pn polynomials in a root of an irreducible congruence of degree n modulo p, a prime.
E. H. Moore90 proved the last theorem and others on finite fields.
K. Zsigmondy36 noted that the number of congruences of degree n modulo p, having no irreducible factor of degree i, is
where I is the number of functions of degree i irreducible modulo p.
G. Cordone91 noted that if a function is prime to each of its derivatives with respect to each prime modulus pi,..., p« and is irreducible modulo M — p^1... pne», it is irreducible with respect to at least one of PI, ..., pn. If F(x) is not identically =0 modulo pi, nor modulo p2, etc., and if it divides a product modulo M and is prime to one factor according to each modulus PI, ..., Pn, then F(x) divides the other factor modulo M.
Let F(x) be a function of degree r irreducible with respect to each prime P!,. .., prt, while/(x) is not divisible by F(x) with respect to any one of the p's, then (pp. 281-8)
\f(x) \ *<*>=! (mod M, F(xfi, <t>r(M) = Mr(l -^-r)... (l - i),
<t>r(M) being the number of functions c1o;r~1+... +cr, in which the c's take . such values 0, 1,..., M—1 whose g. c. d. is prime to M. Let A be the product of these reduced functions modulis M, F(x). Then (pp. 316-8), A= — 1 (mod M, F) if M = pk, 2pk or 4, where p is an odd prime, while As +1 in all other cases.
Borel and Drach92 gave an exposition of the theory of Galois imaginaries from the standpoint of Galois himself.
H. Weber93 considered the finite field (Congruenz Korper) formed of the pn classes of residues modulo p of the polynomials, with integral coefficients, in a root of an irreducible equation of degree n. He proved the generalization of Fermat's theorem, the existence of primitive roots, and the fact that every element is a square or a sum of the squares of two elements.
Ivar Damm94 gave known facts about the roots of congruences modulis P> f(%)) where/(a;) is irreducible modulo p, without exhibiting the second modulus and without making it clear that it is not a question of ordinary congruences modulo p. Let e be a fixed primitive root of the prime p. Then the roots of every irreducible quadratic congruence are of the form a=*= ko, where w2 = e. Let kp+1 = e, ki = kp.
"Bull New York Math. Soc., 3, 1893-4, 73-8.
•°Math. Papers Chicago Congress of 1893, 1896, 208-226; University of Chicago Decennial
Publications, (1), 9, 1904, 7-19. ME1 Progreso Matemdtico, 4, 1894, 265-9. MIntrod. theorie des nombres, 1895, 42-50, 58-62, 343-350. "Lehrbuch der Algebra, II, 1896, 242-50, 259-261; ed. 2, 1899, 302-10, 320-2. MBidrag till Laran om Kongruenser med Primtalsmodyl, Difls., Upsala, 1896, 86 pp.