CHAP, viii] HIGHER CONGEUBNCEB, GALOIS IMAGINABIES. 241 The product of the incongruent primary prime functions modulo p whose degree divides w is congruent modulo p to \T\ = XP*-X. Then, if \f/(p) is the number of primary prime functions of any degree p, SdV'(d) =p*, where the summation extends over all divisors d of w. A comparison of this with ZZV(d)=;pp above shows that N(p)~p\f/(p). Another proof is based on the fact that is congruent modulis p, P to a polynomial in y with integral coefficients which is a prune function. Moreover, if in (3) we associate the linear factors in which the F'a 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 o, 6, . . .) is congruent modulo p to \m\-H\m/db\... H\m/a\-IL\m/dbc\ ... H. J. S. Smith72 gave an exposition of the theory. E. Mathieu,73 in his famous paper on multiply transitive groups, gave without proof the factorization (p. 301; for m = l, p. 275) h(zpmn-z)^tt\(hz)pm(n-l)+(hzym(n~*+ . . . + (hz)pm+hz+a}, a where a ranges over the roots of aplw==a, while hpmn=h; and (p. 302; for m = l,p.280) where ft ranges over the roots of If 12 is a root of a congruence of degree n whose coefficients are roots of zpm=z and whose first member is prime to zpm—z, then (p. 303) all the roots of zpmn=z are given by A0+A1£l+ . . . +An_ii2n~1, where the A's satisfy zpm=z. J. A. Serret,74 in contrast to his70 earlier 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 /*. A simple "British Assoc. Reports, 1860, 120, §§69-71; Coll. M. Papers, 1, 149-155. "Jour, de Math&natiques, (2), 6, 1861, 241-323. T4Mem. Ac. Sc. de 1'Institut de France, 35, 1866, 617-688. Same in Cours d'algebre sup6-rieure, ed. 4, vol. 2, 1879, 122-189; ed. 5, 1885.