244 History of the Theory of Numbers. [Chap, viii 
[Proof in Pellet.®^] In particular, if p is a primitive root of a prime n, 
we have the irreducible function, modulo p, 
(xP-a;)"-l 
x^'-x-l 
C. Jordan^^ listed irreducible functions [errata, Dickson,^"'^ p. 44]. 
J. A. Serret'^^ determined the product F„ of all functions of degree p" 
irreducible modulo p, a prime. In the expansion of (^ — 1)" replace each 
power ^^" by x" ; denote the resulting polynomial in x by X^. Then 
X.,rn^\{^-irl=ie"'-^y, X.m^X^^'^-X (mod p). 
Hence Vn = Xpr^/Xpn-i. Moreover, 
X,+i = (^-l)''+^ = $a-ir-(^-l)''^Z/-X, (modp). 
Multiply this by the relations obtained by replacing /ibyju+1,- • .,iJi+v — l. 
Thus 
X,+.^Z,(X/-i-l)(X,^;J -1) . ..{X:+l,-l) (mod p). 
Take /i = p"~\ /x+i' = p". Hence 
F„^' "ff" A (mod p), A = Xjnii+x-i-l. 
X=l 
Each /x decomposes into p — 1 factors X—g where ^ = 1,..., p — 1. The 
irreducible functions of degree p" whose product is A are said to belong to 
the Xth class. When x is replaced by x^—x, X^ is replaced by X^+i since ^' 
is replaced by ^'(^ — 1) and hence (^ — 1)" by (^ — l)""*"^; thus A is replaced 
by A+i) while the last factor in F„=nA is replaced by Xpn —1, which is 
the first factor in Vn+i- Hence if F{x) is of degree p" and is irreducible 
modulo p and belongs to the Xth class, F{x^—x) is irreducible or the product 
of p irreducible functions of degree p" according as X= or <p'*— p"""\ 
For n = l, the irreducible functions of the Xth class have as roots poly- 
nomials of degree X in a root of i^ — i=l, which is irreducible modulo p. 
Hence if we eliminate i between the latter and f{i) = x, where f{i) is the 
general polynomial of degree X in i, we obtain the general irreducible 
function of degree p of the Xth class. 
For any n, the determination of the irreducible functions of degree p" of 
the first class is made to depend upon a problem of elimination (Algebre, 
p. 205) and the relation to these of the functions of the Xth class, X>1, is 
investigated. 
G. Bellavitis'^" tabulated the indices of Galois imaginaries of order 2 
for each prime modulus p = 4n+3^63. 
Th. Pepin^" proved that x^ — ny^=l (mod p) has p + 1 sets of solutions 
'Kllomptes Rendus Paris, 72, 1871, 283-290. 
"Jour, de Mathdmatiques, (2), 18, 1873, 301^, 437-451. Same as in Cours d'alg^bre sup^rieure, 
ed. 4, vol. 2, 1879, 190-211. 
"f-Atti Accad. Lincei, Mem. Sc. Fis. Mat., (3), 1, 1876-7, 778-800. 
««Atti Accad. Pont. Nuovi Lincei, 31, 1877-8, 43-52. 
