CHAP. VIII] HlGHEB CONGRUENCES, GALOIS IMAGIN ABIES. 243 Harald Schiitz75 considered a congruence Xn+a1Xn-1+ . . . +an=0 (mod M(x)) in which the a's and the coefficients of M are any complex integers (cf. Cauchy,67 for real coefficients). Let 0,1,. . ., an be the roots of the corresponding algebraic equation. Let Af = 0 have the distinct roots jUi, . . . , jum. Then the congruence has nm distinct roots. For, let X~ap =fi(x) have the factor x— /*;, for i=l, . . ., m. Taking i>l, we have Set x = jut. Then the right member must vanish. Using these and fi (MI) = 0 , we have m independent linear relations for the coefficients of A (a;). C. Jordan76 followed Galois in employing from the outset a symbol for an imaginary root of an irreducible congruence, proved the theorems of Galois, and that, if j, jiy . . . are roots of irreducible congruences of degrees ptt, (f, . . . where p, g, . . . are distinct primes, their product jji ... is a root of an irreducible congruence of degree p V3 .... A. E. Pellet77 stated that, if i is a root of an irreducible congruence of degree v modulo p, a prime, the number of irreducible congruences of degree PI whose coefficients are polynomials in i is tf <?i> • • • ) Qm are the distinct primes dividing PI. Of these congruences, <t>(n)/vi belong to the exponent n if n is a proper divisor of (p*)"1 — 1. Any irreducible function of degree /z modulo p with integral coefficients is a product of 6 irreducible factors of degree ju/5 with coefficients rational in i, where 5 is the g. c. d. of ju> v. In an irreducible function of degree i>i and belonging to the exponent n and having as coefficients rational functions of i, replace x by o;x, where X contains only prime factors dividing n; the resulting function is a product of 2*~~1D/n irreducible functions of degree Xni/1/(2A""1Z)) belonging to the exponent An, where D is the g. c. d. of Xn and p""1— 1, and 2fc~1 is the highest power of 2 dividing the numerators of each of the fractions (p"I/lH-l)/2 and Xn/(2D) when reduced to their lowest terms. Let g be a rational function of it and m the number of distinct values among g, gp, gp2, .... If neither g+gp+ . . . +gpm~l nor v/m is divisible by p, then xp — x — g is irreducible; in the contrary case it is a product of linear functions. Hence if we replace x by xp— x in an irreducible function of degree /* having as coefficients rational functions of i, we get a new irreducible function provided the coefficient of x"~l in the given function is not zero. 7BUntersuchungen tiber Functional Congruenzen, Dies. Gottingen, Frankfurt, 1867. 7BTrait6 des substitutions, 1870, 14-18. "Comptes Rendus Paris, 70, 1870, 328-330.