84 HlSTOBY OF THE THEOEY OF NUMBERS. [CHAP. HI G. Frattini153 noted that, if JP(a, £,. . .) is a homogeneous symmetric polynomial, of degree g with integral coefficients, in the integers a, 0,.'. . less than ra and prime to ra, and if F is prime to ra, then Afe 1 (mod ra) for every integer k prime to m. In fact, F(o, 0,. . .)=F(ka, fcft. . Ossfc^a, 0,. . .) (mod ra). Taking F to be the product ajS. . ., we have Euler's theorem. Another corollary is l)! (modp), for p a prime, which implies Wilson's theorem. *J. L. Wildschtitz-Jessen154 gave an historical account of Fermat's and Wilson's theorems. E. PiccioH155 repeated the work of Dirichlet.40 THE GENERALIZATION F(a, JV)=0 (MOD N) OF FERMAT'S THEOREM. C. F. Gauss160 noted that, if N^pf. . .p," (p's distinct primes), t-l i<j »</<* is divisible by N when a is a prime, the quotient being the number of irreducible congruences modulo a of degree N and highest coefficient unity. He proved that (1) aN where d ranges over all the divisors of JV, and stated that this relation read- ily leads to the above expression for F (a, AT) . [See Ch. XIX on inversion.] Th. Schonemann161 gave the generalization that if a is a power pn of a prime, the number of congruences of degree 2V irreducible in the Galois field of order a is #-^(0, AT). An account of the last two papers and later ones on irreducible congruences will be given in Ch. VIII. J. A. Serret162 stated that, for any integers a and N, F(a, N) is divisible by N. For N~p*, p a prime, this implies that a<Kpc)==l (modpe), when a is prime to p} a case of Euler's theorem. S. Kantor163 showed that the number of cyclic groups of order N in any birational transformation of order a in the plane is N~1F(a) N) . He obtained (1) and then the expression for F(a, N) by a lengthy method completed for special cases. 1BSPeriodico di Mat., 29, 1913, 49-53. 1MNyt Tidsskrift for Mat., 25, A, 1914, 1-24, 49-68 (Danish). 165Periodico di Mat., 32, 1917, 132-4. ""Posthumous paper, Werke, 2, 1863, 222; Gauss-Maser, 611. m Jour, fto Math.., 31, 1846, 269-325. Progr. Brandenburg, 1844. »3Nouv. Ann. Math., 14, 1855, 261-2. 1MAnnali di Mat., (2), 10, 1880, 64r-73. Comptes Rendua Paris, 96, 1883, 1423.