Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 235 
If X has been expressed as a product of relatively prime factors modulo 
p, we can express X as a product of a like number of factors mod p" con- 
gruent to the former factors modulo p. There is a fragment on the case 
of multiple factors. 
C. G. J. Jacobi" noted that, if g is a prime 6n — 1, x*+^=l (mod q) has 
q — 1 imaginary roots a + 6V— 3, where a +36^=1 (modg), besides the roots 
±1. 
E. Galois®^ employed imaginary roots of any irreducible congruence 
F(a;) = (mod p), where p is a prime. Let i be one imaginary root of this 
congruence of degree v. Let a be one of the p" — 1 expressions 
a-\-aii+a2i^-\- . . . +a^-ii''~^ 
in which the a's are integers <p, not all zero. Since each power of a can 
be expressed as such a polynomial, we have a" = 1 for some positive integer 
n. Let n be a minimum. Then 1, a, . . . , a"~^ are distinct. Multiply them 
by a new polynomial (3 ini; we get n products distinct from each other and 
from the preceding powers of a. If 2n<p'' — 1, we use a new multiplier, 
etc. Hence n divides p" — 1, and 
(2) 0^"-^ = !. 
[This is known as Galois's generalization of Fermat's theorem.] It follows 
that there exist primitive roots a such that a^p^l if e<p'' — L Any primi- 
tive root satisfies a congruence of degree v irreducible modulo p. 
Every irreducible function F{x) of degree v divides x^''—x modulo p. 
Since jF(x)[^"=F(xO modulo p, the roots of F{x)=0 are 
All the roots of x^' = x are polynomials in a certain root ^, which satisfies 
an irreducible congruence of degree v. To find all irreducible congruences 
of degree v modulo p, delete from x^'' — x all factors which it has in common 
with x^^—x, iJL<v. The resulting congruence is the product of the desired 
ones; the factors may be obtained by the method of Gauss, since each of 
their roots is expressible in terms of a single root. In practice, we find by 
trial one irreducible congruence of degree v, and then a primitive root of 
(2); this is done for p = 7, v = 3. 
Any congruence of degree n has n real or imaginary roots. To find 
them, we may assume that there is no multiple root. The integral roots 
are found from the g. c. d. of F{x) and x^~^ — l. The imaginary roots of 
the second degree are found from the g. c. d. oi F{x) and x^'~^ — l; etc. 
V. A. Lebesgue^^ noted that, if p is a prime, the roots of all quadratic 
"Jour, fiir Math., 2, 1827, 67; Werke, 6, 235. 
"Sur la tMorie des nombres, Bulletin des Sciences Mathlmatiques de M. Ferussac, 13, 1830, 428. 
Reprinted in Jour, de Math^matiques, 11, 1846, 381; Oeuvres Math. d'Evariste Galois, 
Paris, 1897, 15-23; Abhand. Alg. Gleich. Abel u. Galois, Maser, 1889, 100. 
"Jour, de Math^matiques, 4, 1839, 9-12. 
