252 History of the Theory of Numbers. [Chap, viii 
the residues prime to p, (t>{x) constitute the direct product of the groups 
with respect to the various p, 0.(a:). Not every abelian group can be repre- 
sented as a congruence group composed of a complete set of prime residues 
with respect to Fj, . . ., Fx, where the F's are functions of a single variable. 
Mildred Sanderson^ ^^ employed two moduU m and P{y), the first being 
any integer and the second any polynomial in y with integral coefficients. 
Such a polynomial /(?/) is said to have an inverse /i (y) if //i= 1 (mod m, P). 
If P{y) is of degree r and is irreducible with respect to each prime factor of 
m, a function f{y), whose degree is <r, has an inverse moduhs m, P{y), if 
and only if the g. c. d. of the coefficients of /(?/) is prune to m. For such 
an/, /" = 1 (mod vi, P), where n is Jordan's function Jr{fn) [Jordan, ^"^ 
Ch. V]. In case m is a prime, this result becomes Galois'^^ generalization 
of Fermat's theorem. The product of the n distinct residues having 
inverses moduHs m, P{y), is congruent to —1 when m is a power of an odd 
prime or the double of such a power or when r= 1, w = 4; but congruent to 
+ 1 in all other cases — a two-fold generahzation of Wilson's theorem. 
There exists a polynomial P{y) of degree r which is irreducible with respect 
to each prime factor of m. Then if A{y), B{y) are of degrees <r and their 
coefficients are not all divisible by a factor of m, there exist polynomials 
a(i/), ^{y), such that aA+/3B=l (mod m, P). 
Several writers^^^ discussed the irreducible quadratic factors modulo p 
of {x'^—\)/{x^ — l), where A' = 1 or 2, p is a prime, a a divisor of p-fl. 
G. Tarry^^^ noted that, if f=q (mod m), where 5 is a quadratic non- 
residue of the prune m, the Galois imaginary a+hj is a primitive root if 
its norm {a-\-'bj){a — hj) is a primitive root of m and if the ratio a:h and the 
analogous ratios of the coordinates of the first m powers of a-\-hj are incon- 
gruent. 
L. E. Dickson^ ^^ proved that two polynomials in two variables with 
integral coefficients have a unique g. c. d. modulo p, a prime. Thus the 
unique factorization theorem holds. 
G. Tarr>'^^^ stated that Ap is a primitive root of the GF[p^] if the norm 
of A = a-\-'bj is a primitive root of p and if the imaginar>^ p belongs to the 
exponent p+1. The 0(p-f 1) numbers p are found by the usual process 
to obtain the primitive roots of a prime. 
U. Scarpis^-° proved that an equation of degree v irreducible in the 
Galois field of order p" has in the field of order p"*" either v roots or no 
root according as v is or is not a di\dsor of m [Dickson^°^, p. 19, lines 7-9]. 
Cubic Congruences. 
A. Cauchy^^° solved y'^-\-By-\-C={) (mod p) when it has three distinct 
'"Annals of Math., (2), 13, 1911, 36-9. 
"•L'interm^diaire dea math., 18, 1911, 195, 246; 19, 1912, 61-69, 95-6; 21, 1914, 158-161; 22, 
1915, 77-8. Sphinx-Oedipe, 7. 1912, 2-3. 
"'Assoc, fran^. av. sc, 40, 1911, 12-24. "sBuU. Amer. Math. Soc, (2), 17, 1911, 293-4. 
"'Sphinx-Oedipe, 7, 1912, 43^, 49-50. ""AnnaU di Mat., (3), 23, 1914, 45. 
""Exercices de Math., 4, 1829, 279-292; Oeuvres, (2), 9, 326-333. 
