Chap. VIII] HIGHER CONGRUENCES, GaLOIS ImAGINARIES. 251 
W. H. Bussey^°^ gave for each Galois field of order < 1000 companion 
tables showing the residues of the successive powers of a primitive root, 
and the powers corresponding to the residues arranged in a natural order. 
These tables serve the same purposes in computations with Galois fields 
that tables of indices serve in computations with integers modulo p", where 
p is a prime. 
G. Voronoi^°^ proved the theorem of Stickelberger.^^ Thus, for n = 3, 
(D/p) = — 1 only when v = 2. Hence a cubic congruence has a single root 
if (D/p) = —1, and three real roots or none if {D/p) = +1. 
P. Bachmann^^° developed the general theory from the standpoint of 
Kronecker's modular systems and considered its relation to ideals (p. 241). 
M. Bauer^^^ employed a polynomial /(z) of degree n irreducible modulo 
p, and another one M{z) of degree less than that of f{z) and not divisible 
by/(2) modulo p. Then if {t, a) = l, the equation 
/(2)+p«M(2)=0 
is irreducible. The case a = 1 is due to Schonemann^^ (p. 101). 
G. Arnoux,^^^ starting with any prime m and integer n, introduced a 
symbol i such that i^^^ 1 (mod m) and such that i, i^, . . ., i^ are distinct, 
where s = m" — 1, without attempting a logical foundation. If /(x) is irre- 
ducible modulo m and of degree n, there is only a finite number of distinct 
residues of powers of x modulis/(x), m] let x^ andx'^'^^have the same residue. 
Thus x^ — 1 is divisible by/(x) modulo m. It is stated (p. 95) without proof 
that p divides s. "Call a a root of /(a:) = 0. To make a coincide with the 
primitive root i of a;^= 1, we must take p = s, whence every such primitive 
root is a root of an irreducible congruence of degree n modulo m." Follow- 
ing this inadequate basis is an exposition (pp. 117-136) of known properties 
of Galois imaginaries. 
L. I. Neikirk^^^ represented geometrically the elements of the Galois 
field of order p" defined by an irreducible congruence 
/(x) =rc"+aia:"-^H- . . . -fa„=0 (mod p). 
Let j be a root of the equation f{x) = and represent 
Ci/~^+ . . . +c„_ii-Fc„ (c's integers) 
by a point in the complex plane. The p"" points for which the c's are chosen 
from 0, 1, . . ., p — 1 represent the elements of the Galois field. 
G. A. Miller^^^ listed all possible modular systems p, 4>{x), where p is a 
prime and the coefficient of the highest power of x is unity, in regard to 
which a complete set of prime residues forms a group of order ^12. If 
4>{x) is the product of k distinct irreducible functions 4>i, . . . , (f)^ modulo p, 
"SBuU. Amer. Math. Soc, 12, 1905, 21-38; 16, 1909-10, 188-206. 
lO'Verhand. III. Internat. Math. Kongress, 1905, 186-9. 
""AUgemeine Arith. d. Zahlenkorper, 1905, 81-111. 
"iJour. fur Math., 128, 1905, 87-9. 
"'Arithm^tique Graphique, Fonctiona Arith., 1906, 91-5. 
"'BuU. Amer. Math. Soc, 14, 1907-8, 323-5. 
"«Archiv Math. Phys., (3), 15, 1909-10, 115-121. 
