Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 239 
G. Eisenstein^^ stated that if /(a;) = is irreducible modulo p, and a is a 
root of the equation /(x) = of degree n, and if ao, ai,. . . are any integers, 
K = ao-\-aia-\-. . . . +a„_ia"~-^ 
is congruent modulis p, a to one and but one expression 
5 = 60/3+61^^+62/3^'+ . . . +6„_i/3^"-\ 
where the 6's are integers and /5 is a suitably chosen function of a. Hence 
the p" numbers B form a complete set of residues modulis p, a. If co is a 
primitive nth root of unity, and if 
(/)(X) =a+ajV+co2V'+ . . . +cu^"-i^V""\ 
the product 0(X)0(X') ... is independent of a if X+X'+ ... is divisible by n. 
Th. Schonemann^^ proved the last statement in case n is not divisible 
by p. To make K = B, raise it to the powers p, p^,. . ., p"~^ and reduce 
by j8^"=/3 (mod p, a). This system of n congruences determines /3 uniquely 
if the cyclic determinant of order n with the elements hi is not divisible by 
p; in the contrary case there may not exist a (3. The statement that the 
expressions B form p" distinct residues is false if jS is a root of a congruence 
of degree <n irreducible modulo p; it is true if /3 is a root of such a con- 
gruence of degree n and if 
i8+/3^+ . . . +/3^""'^0 (mod p, a). 
J. A. Serref^" made use of the g. c. d. process to prove that if an irre- 
ducible 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) = (mod p) of degree v>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 a^''=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 v, has integral 
coefficients, and is irreducible modulo p, there exist polynomials f{x) and 
x(x) with integral coefficients such that 
x''''-x=f{x)F(x) +px{x). 
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 ingeni- 
ous argument; but as he made use of imaginary roots of congruences without 
giving an adequate basis to their theory, the proof is not conclusive. 
'sjour. fur Math., 39, 1850, 182. 
«'Jour. fiir Math., 40, 1850, 185-7. 
"Cours d'algdbre sup6rieure, ed. 2, Paris, 1854, 343-370. 
