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[Proof in Pellet.88]   In particular, if p is a primitive root of a prime n, we have the irreducible function, modulo p,
C. Jordan78 listed irreducible functions [errata, Dickson,102 p. 44].
J. A. Serret79 determined the product Vn of all functions of degree pn irreducible modulo p, a prune. In the expansion of (-!)" replace each power * by x^\ denote the resulting polynomial in x by ZM. Then
-* (mod
Hence Vn*=X9*/X9-i.   Moreover,
Multiply this by the relations obtained by replacing ju by M+!> . . . , n+v1. Thus
x^ex/z^-iXXKl -i) . . .(XK^-I) (mod p).
Take M = pn~"1, /A + v = pn.   Hence "n
Each /x decomposes into p  1 factors X g where 0=1,.. ., p 1. The irreducible functions of degree pn whose product is /x are said to belong to the Xth class. When x is replaced by xp x, X^ is replaced by XM+1 since ' is replaced by (1) and hence (-1)" by ( I)"*1; thus/x is replaced by A+i, while the last factor in Fn=H/x is replaced by Xppl 1, which is the first factor in Fw+1. Hence if F(x] is of degree pn and is irreducible modulo p and belongs to the Xth class, F(xp x) is irreducible or the product of p irreducible functions of degree pn according as X= or <pn p71"1.
For n= 1, the irreducible functions of the Xth class have as roots polynomials of degree X in a root of ipi=l, which is irreducible modulo p. Hence if we eliminate i between the latter and/(^)=x, where /(i) is the general polynomial of degree X in i, we obv%in the general irreducible function of degree p of the Xth class.
For any n, the determination of the irreducible functions of degree pn of the first class is made to depend upon a problem of elimination (Alg&bre, p. 205) and the relation to these of the functions of the Xth class, X>1, is investigated.
G. Bellavitis79tt tabulated the indices of Galois imaginaries of order 2 for each prime modulus p = 4n-|~3^63.
Th. Pepin80 proved that x'2  ny2=l (mod p) has p+1 sets of solutions
'ComptH RcnduB Paris, 72, 1871, 283-290.
"Jour, do MtttMmatiques, (2), 18, 1873, 301-4, 437-451.   Same as in Cours d'alg&re aup&ieure,
wl. 4, vol. 2, 1879, 190-211.
'"Atti Accacl. Lincei, Mem. Sc. Fis. Mat., (3), 1, 1876-7, 778-800. i0Atti Accad. Pont. Nuovi Lincei, 31, 1877-8, 43-52.