Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 247 
E. H. Moore^^ stated that every finite field (Korper) is, apart from nota- 
tions, a Galois field composed of the p" polynomials in a root of an irreducible 
congruence of degree n modulo p, a prime. 
E. H. Moore^° proved the last theorem and others on finite fields. 
K. Zsigmondy^^ noted that the number of congruences of degree n 
modulo p, having no irreducible factor of degree i, is 
P"-({)p"-^+(2)p"-'' 
where / is the number of functions of degree i irreducible modulo p. 
G. Cordone^^ noted that if a function is prime to each of its derivatives 
with respect to each prime modulus Pi,..., Pn and is irreducible modulo 
M = piK . .pn", 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 
Pi, . . ., Pn, while /(x) is not divisible by F{x) with respect to any one of the 
p's, then (pp. 281-8) 
l/(x)[-^^^^^l (mod M, F(x)), 0,(M)=M'-(l-^,y . .(l-^), 
<}>r{M) being the number of functions Cix''~^+ . . . -\-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 = p^, 2p^ or 4, where p is an odd prime, while 
A= + l in all other cases. 
Borel and Drach^^ gave an exposition of the theory of Galois imaginaries 
from the standpoint of Galois himself. 
H. Weber^^ considered the finite field (Congruenz Korper) formed of the 
p" classes of residues modulo p of the polynomials, with integral coefficients, 
in a root of an irreducible equation of degree n. He proved the generaliza- 
tion 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 Damm^* gave known facts about the roots of congruences modulis 
p, /(x), where /(x) 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± hoi, where co^ = e. Let k^^^ = e, ki = k^. 
89Bull. 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. 
"El Progreso Matemdtico, 4, 1894, 265-9. 
"Introd. th^orie 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. 
"Bidrag till Laran om Kongruenser med Primtalsmodyl, Diss., Upsala, 1896, 86 pp. 
