84 History of the Theory of Numbers. [Chap, hi 
G. Frattini^^ noted that, if F{a, j3, . . . ) is a homogeneous symmetric 
poljTiomial, of degree g with integral coefficients, in the integers a, /3, . . . 
less than m and prime to m, and if F is prime to 7??, then k°= 1 (mod m) for 
ever>' integer A- prime to m. In fact, 
F(a, jS,. . .) = F{ka, k^,.. .) = k^F{a, 0,...) (mod m). 
Taking F to be the product a^. . ., we have Euler's theorem. Another 
corollary is 
u\l+j)=l + (p-l)l (modp), 
for p a prime, which implies Wilson's theorem. 
*J. L. Wildschlitz-Jessen^^^ gave an historical account of Fermat's and 
Wilson's theorems. 
E. Piccioh^" repeated the work of Dirichlet.'*° 
The Generalization F{a,N) = (mod N) of Fermat's Theorem. 
C. F. Gauss^^° noted that, if N=pi^ . . .p/* (p's distinct primes), 
»=1 »■<;■ x<i<k 
is divisible by N when a is a prime, the quotient being the number of irre- 
ducible congruences modulo a of degree N and highest coefficient unity. 
He proved that 
(1) a^=2F(a, d), F{a,\)=a, 
where d ranges over all the divisors of N, and stated that this relation read- 
ily leads to the above expression for F (a, N). [See Ch. XIX on inversion.] 
Th. Schonemann^^^ gave the generalization that if a is a power p" of a 
prime, the number of congruences of degree A^ irreducible in the Galois field 
of order a is N~'^F{a, N). 
An account of the last two papers and later ones on irreducible con- 
gruences will be given in Ch. VIII. 
J. A. Serret^^^ stated that, for any integers a and iV, F{a, N) is divisible 
by N. For N=p\ p a prime, this implies that 
a</>(pO = l (modpO, 
when a is prime to p, a case of Euler's theorem. 
S. Kantor^^^ showed that the number of cycHc groups of order N in any 
birational transformation of order a in the plane is N~^F{a, N) . He obtained 
(1) and then the expression for F(a, N) by a lengthy method completed for 
special cases. 
iwPeriodico di Mat., 29, 1913, 49-53. 
i^Nyt Tidsskrift for Mat., 25, A, 1914, 1-24, 49-68 (Danish). 
i"Periodico di Mat., 32, 1917, 132-4. 
"oPosthumous paper, Werke, 2, 1863, 222; Gauss-Maser, 611. 
"iJour. fiir Math., 31, 1846, 269-325. Progr. Brandenburg, 1844. 
i"Nouv. Ann. Math., 14, 1855, 261-2. 
"'AnnaU di Mat., (2), 10, 1880, 64-73. Comptes Rendus Paris, 96, 1883, 1423. 
