■^ 
Chap. VIII] HiGHER CONGRUENCES, GaLOIS ImAGINARIES. 241 
The product of the incongruent primary prime functions modulo p 
whose degree divides tt is congruent modulo p to 
Then, if xpip) is the number of primary prime functions of any degree p, 
l!id\l/{d)=p', where the summation extends over all divisors d of tt. A com- 
parison of this with XN{d)=p'' above shows that N{p)=p\l/{p). Another 
proof is based on the fact that 
(y-A){y-A')...{y-A''-') 
is congruent modulis p, P to a polynomial in y with integral coefficients 
which is a prime function. Moreover, if in (3) we associate the linear 
factors in which the F's belong to the same exponent, we obtain a factor 
of the left member which is irreducible modulo p. 
The product of the incongruent primary prime functions of degree m 
(m being divisible by no primes other than a, 6, . . . ) is congruent modulo p 
to 
\m\-'n.\m/ab\ . . . 
Il\m/a\'Il\m/abc\ 
H. J. S. Smith^^ gave an exposition of the theory. 
E. Mathieu,^^ in his famous paper on multiply transitive groups, gave 
without proof the factorization (p. 301; for m = l, p. 275) 
h{z^"'''-z)=u\(hzy"'^''-''+ihzy'"^''-''+ . ..+(hzy"'+hz+a}, 
a 
where a ranges over the roots of a^'^^a, while /i^"*"=/i; and (p. 302; for 
m = l,p. 280) 
/^(gp'"" - z) =n(;i^ V" -hz-^), 
where jS ranges over the roots of 
If 12 is a root of a congruence of degree n whose coefficients are roots of 
z'^"'=z and whose first member is prime to z^'^ — z, then (p. 303) all the roots 
of z^""'=z are given by ^o+^i^+- • .+^„-il2''~\ where the A's satisfy 
Z^ =Z. 
J. A. Serret,'^^ in contrast to his^° earher exposition, here avoided at 
the outset the use of Galois imaginaries. An irreducible function of degree 
V modulo p divides x^—x modulo p if and only if v divides ^t. A simple 
"British Assoc. Reports, 1860, 120, §§69-71; Coll. M. Papers, 1, 149-155. 
"Jour, de Math6matiques, (2), 6, 1861, 241-323. 
"M6m. Ac. Sc. de I'Institut de France, 35, 1866, 617-688. Same in Cours d'algfebre sup6- 
rieure, ed. 4, vol. 2, 1879, 122-189; ed. 5, 1885. 
