CHAP, viii] HIGHER CONGRUENCES, GALOIS IMAGINARIES. 239 G. Eisenstein68 stated that if f(x)=Q is irreducible modulo p, and a is a root of the equation /(z) = 0 of degree n, and if a0, alt . . . are any integers, is congruent modulis p, a to one and but one expression where the bjs are integers and ft is a suitably chosen function of a. Hence the pn numbers B form a complete set of residues modulis p, a. If co is a primitive nth root of unity, and if <KX) = a+coV+o>2V2+ . . . +co(n-1)Vn~1, the product 0(X)0(X') ... is independent of a if X+X'H- ... is divisible by n. Th. Schonemann69 proved the last statement in case n is not divisible by p. To make K=B, raise it to the powers p, p2, . . ., pn~l and reduce by /3pn=/3 (mod p, a). This system of n congruences determines /3 uniquely if the cyclic determinant of order n with the elements bt is not divisible by p; in the contrary case there may not exist a /5. The statement that the expressions B form pn distinct residues is false if /3 is a root of a congruence of degree <n irreducible modulo p; it is true if ft is a root of such a congruence of degree n and if 0+jS*+ . . . +/3pn~Vo (mod p, a). J. A. Serret70 made use of the g. c. d. process to prove that if an irreducible function F(x) divides a product modulo p, a prime, it divides one factor modulo p. Then, following Galois, he introduced an imaginary quantity i verifying the congruence F(i) = 0 (mod p) of degree z>>l, but gave no formal justification of their use, such as he gave in his later writings. However, he recognized the interpretation that may be given to results obtained from their use. For example, after proving that any polynomial a(i) with integral coefficients is a root of ap"=a (mod p), he noted that this result, for the case a = i, may be translated into the following theorem, free from the consideration of imaginaries: If F(x) is of degree *>, has integral coefficients, and is irreducible modulo p, there exist polynomials f(x) and x(x) with integral coefficients such that The existence of an irreducible congruence of any given degree and any prime modulus is called the chief theorem of the subject. After remarking that Galois had given no satisfactory proof, Serret gave a simple and ingenious argument ; but as he ma'de use of imaginary roots of congruences without giving an adequate basis to their theory, the proof is not conclusive. 88Jour. fur Math., 39, 1850, 182. •Mour. fur Math., 40, 1850, 185-7. 70CouT8 d'alg&bre sup6rieure, ed. 2, Paris, 1854, 343-370.