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 A?/ be of order n and irreducible modulo p. Then Ay is con- 
gruent modulis p, A?/ to one and but one of the p" forms 
(4) s'c,^ (c, = 0, l,...,p-l). 
If It is any one of these forms (4) and if e = 'p^ — \, Guldberg stated the 
analogue of Fermat's theorem 
dfu 
^='M (mod p, ^y), 
but incorrectly gave the right member to be unity [cf. Epsteen/''^, Dickson^"^]. 
L. Stickelberger^^ considered F{x) =x^+aix''~^-\- . . . with integral coeffi- 
cients, 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 
prime ideals that 
(f)=(-i)»-. 
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[p''] of order 
p" by induction from r = n to r = qn, by showing that 
(a:^"'-a:)/(x^"-x) 
is a product of factors of degree q belonging to and irreducible in the 
(?F[p"]. Any such factor defines the GF[p'"']. 
L. Kronecker^°° treated congruences modulis p, P{x) from the stand- 
point of modular systems. 
F. S. Carey ^°^ gave for each prime p< 100 a table of the residues of the 
first p + 1 powers of a primitive root a-\-hj of z^~^=l (mod p) where /=»' 
(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 x^—v. Methods of "solving" 
2?"-!=! are discussed. In particular, for n = 3, there is given a primitive 
root for each prime p< 100. 
L. E. Dickson^"^ gave a systematic introductory exposition of the theory, 
with generalizations and extensions. 
M. Bauer^"^ proved that, if /(a;) =0 is an irreducible equation with inte- 
gral coefficients and leading coefficient unity, w a root, D its discriminant, 
d = D/k^ that of the domain defined by w, p sl prime not dividing k, x>l, 
«8Verhand. I. Internat. Math. Kongress, 1897, 186. 
"^BuU. Amer. Math. Soc, 6, 1900, 203-4. 
looVorlesungen iiber Zahlentheorie, I, 1901, 212-225 (expanded by Hensel, p. 506). 
i"Proc. London Math. Soc, 33, 1900-1, 294-310. 
lo'Linear groups with an exposition of the Galois field theory, Leipzig, 1901, pp. 1-71. 
lo^Math. Naturw. Berichte aus Ungarn, 20, 1902, 39-42; Math. 6s Phyp. Lapok, 10, 1902, 28-33. 
