CHAP. VIII] HlGHEK CONGRUENCES, GALOIS IMAGINABIES. 251 W. H. Bussey108 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 pn, where p is a prune. G. Voronoi109 proved the theorem of Stickelberger.98 Thus, for n = 3, (D/p) = — 1 only when *> = 2. Hence a cubic congruence has a single root if (D/p) = — 1, and three real roots or none if (D/p) = +1. P. Bachmann110 developed the general theory from the standpoint of Kronecker's modular systems and considered its relation to ideals (p. 241). M. Bauer111 employed a polynomial f(z) of degree n irreducible modulo p, and another one M(z) of degree less than that of f(z) and not divisible by f(z) modulo p. Then if (t, a) = 1, the equation is irreducible. The case a = l is due to Schonemann65 (p. 101). G. Arnoux,112 starting with any prune m and integer n, introduced a symbol i such that t*= 1 (mod m) and such that it i2, . . . , i* are distinct, where s = wn— 1, without attempting a logical foundation. If f(x) is irreducible modulo m and of degree n, there is only a finite number of distinct residues of powers of x modulis/(z), m; let xk and xk+p have the same residue. Thus xp — 1 is divisible byf(x) modulo m. It is stated (p. 95) without proof that p divides s. "Call a a root of /(x)=0. To make a coincide with the primitive root i of xs=l, we must take p = s, whence every such primitive root is a root of an irreducible congruence of degree n modulo m." Following this inadequate basis is an exposition (pp. 117-136) of known properties of Galois imaginaries. L. I. Neikirk113 represented geometrically the elements of the Galois field of order pn defined by an irreducible congruence f(x) = xn+a,xn~l+ . . .+an=Q (mod p). Let j be a root of the equation f(x) = 0 and represent Cij*~l+ - - - +£n-ij+cn (c's integers) by a point in the complex plane. The pn points for which the c's are chosen from 0, 1, . . ., p — 1 represent the elements of the Galois field. G. A. Miller114 listed all possible modular systems p, <l>(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 g 12. If <j>(x) is the product of k distinct irreducible functions <£i, ...,<£* modulo p, ""Bull. Amer. Math. Soc., 12, 1905, 21-38; 16, 1909-10, 188-206. "'Verhand. III. Internat. Math. Kongress, 1905, 186-9. u°Allgemeine Arith. d. Zahlenkorper, 1905, 81-111. "'Jour, fiir Math., 128, 1905, 87-9. UIArithm6tique Graphique, Fonctions Arith., 1906, 91-5. "'Bull. Amer. Math. Soc., 14, 1907-8, 323-5. '"Archiv Math. Phys., (3), 15, 1909-10, 115-121.