Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 243 
Harald Schiitz^^ considered a congruence 
Z'^+aiX"-i+ . . . +a„=0 (mod Mix)) 
in which the a's and the coefficients of M are any complex integers 
(cf. Cauchy,®^ for real coefficients). Let ai,..., a„ be the roots of the 
corresponding algebraic equation. Let M = have the distinct roots 
III,. . ., jirn- Then the congruence has n"* distinct roots. For, let X — a^ 
=fi{x) have the factor x—jii, for i = 1, . . ., m. Taking i>l, we have 
fi{^)=fl{^)+0.p-(lpi' 
Set X = 111. Then the right member must vanish. Using these and /i (/^i) = , 
we have m independent linear relations for the coefficients of /i(x). 
C. Jordan^® followed Galois in employing from the outset a symbol for 
an imaginary root of an irreducible congruence, proved the theorems of 
Galois, and that, if j, ji, . . . are roots of irreducible congruences of degrees 
p", q^,. . . where p, q, . . . are distinct primes, their product jji ... is a root 
of an irreducible congruence of degree p^q^ .... 
A. E. Pellet^^ stated that, if t is a root of an irreducible congruence of 
degree v modulo p, a prime, the number of irreducible congruences of degree 
Vi whose coefficients are polynomials in i is 
— jp""! — Sp'"'i/5i+Sp'"'i/9i«2— ...+( — l)"'p''V9i- ■ -Sm } 
if qi,..., qm are the distinct primes dividing vi. Of these congruences, 
4){n)/vi belong to the exponent n if n is a proper divisor of (p")"' — L 
Any irreducible function of degree ix modulo p with integral coefficients 
is a product of 5 irreducible factors of degree ix/b with coefficients rational 
in i, where b is the g. c. d. of fx, v. 
In an irreducible function of degree vi and belonging to the exponent n 
and having as coefficients rational functions of i, replace x by x^, where X 
contains only prime factors dividing n; the resulting function is a product of 
2^~^D/n irreducible functions of degree \nvi/{2^~^D) belonging to the 
exponent \n, where D is the g. c. d. of \n and p""* — 1, and 2^~^ is the highest 
power of 2 dividing the numerators of each of the fractions (p'"''+l)/2 and 
Xn/(2Z)) when reduced to their lowest terms. 
Let gf be a rational function of i, and m the number of distinct values 
among g, g^, g^ ,. . .. If neither g-{-g^-{- . . . +9'^"*" nor v/m is divisible by 
p, then x^ — x — g is irreducible; in the contrary case it is a product of 
linear functions. 
Hence if we replace a; by x^ — x in an irreducible function of degree /x 
having as coefficients rational functions of i, we get a new irreducible 
function provided the coefficient of x''~^ in the given function is not zero. 
^^Untersuchungen liber Functionale Congruenzen, Diss. Gottingen, Frankfurt, 1867. 
^«Trait^ des substitutions, 1870, 14-18. 
"Comptes Rendus Paris, 70, 1870, 328-330. 
