246 History of the Theory of Numbers. [Chap, viii 
the differences of the roots of /(x") = is a quadratic non-residue or residue 
of p [Pellet^^j. Let Ai be the like product for J{x). Then A = (-l)'2-' 
/(0)Ai". Hence /(ax- +6) is irreducible if ( — l)7(6)/a'' is a quadratic non- 
residue and then /(ax '+6) is irreducible modulo p for every i and even v. 
0. H. Mitchell^ gave analogues of Fermat's and Wilson's theorems 
moduhs p (a prime) and a function of x. 
A. E. Pellet^ considered the exponent n to which belongs the product P 
of the roots of a congruence F(x) = of degree v irreducible modulo p. 
If g is a prime factor of n, F(x') is irreducible or the product of q irreducible 
factors of degree v modulo p according as q is not or is a divisor of {p — \)/n. 
In particular, F{x^) is irreducible modulo p if, for v even, X contains only- 
prime factors of n not dividing {p — \)/n; for v odd, we can use the factor 2 
in X only once if p = 4mH-l. Let i be a root of F(x) = 0, I'l a root of an 
irreducible congruence Fi(x) = (mod p) of degree v^ prime to v. Then 
ill is a root of an irreducible congruence G(x) = (mod p) of degree vvx. 
F{x) belongs to the exponent Nn modulo p, where n is prime to (p'— 1) 
-^\{p — \)N\. Let qi be a prime factor of .V not dividing p — 1. Then 
G(x'') is irreducible or decomposes into qi irreducible factors of degree vvi 
according as qi is not or is a divisor of (p'' — l)/N. Thus G(x^) is irreducible 
if X contains onlj' prime factors of N dividing neither p — 1 nor {p'' — l)/N. 
0. H. MitchelP^ defined the prime totient of /(x) to mean the number 
of polynomials in x, incongruent modulo p, of degree less than the degree of 
(x) and having no factor in conmion with / modulo p. Those which 
contain S, but no prime factor of / not contained in S, are called >S-totitives 
of/. 
C. Dina^^ proved known results on congruences moduhs p and F{x). 
A. E. Pellet^^ proved that, if ju distinct values are obtained from a 
rational function of x with integral coefficients by replacing x successively 
by the 77i roots of an irreducible congruence modulo p, then ^i is a divisor 
of m and these /jl values are the roots of an irreducible congruence. Thus 
if A is a rational function of any number of roots of congruences irreducible 
modulo p, and p is the number of distinct values among A, A^, A^\. . ., 
these values satisfy an irreducible congruence modulo p. If A belongs to 
the exponent n modulo p, then v is the least positive integer for which p''= 1 
(mod n). He proved a result of Serret's^^ stated in the following form: If, 
in an irreducible function F{x) modulo p of degree v and exponent n, x is 
replaced by x^, where X contains only primes dividing n, then F(x^) is a 
product of irreducible factors of degree vq and exponent n\, where q is the 
least integer for which ^"^=1 (mod n\). He proved the first theorem of 
Pellet^ and the last one of Pellet." 
"Johns Hopkins University Circulars, 1, 1880-1, 132. 
"Comptps Rendus Paris, 93, 1881, 1065-6. Cf. Pellet." 
"Amer. Jour. Math., 4, 1881, 25-38. 
•^Giomale di Mat., 21, 1883, 234-263. For comments on 263-9, see the chapter on quadratic 
reciprocity law. 
««Bull. Soc. Math. France, 17, 1888-9, 156-167. 
