CHAP, viii] HIGHER CONGRUENCES, GALOIS IMAGINABIES. 245 & y selected from 0, 1, . . . , p — 1, provided n is a quadratic non-residue of the prime p. Then x+ yVn is a root of £p+1s=l (mod p), which therefore has 2>-j-l complex roots, all a power of one root. There is a table of indices for these roots when p = 29 and p = 41. [Lebesgue.63] A. E. Pellet81 considered the product A of the squares of the differences of the roots of a congruence f(x) =0 (mod p) having no equal roots. Then A is a quadratic non-residue of p if f(x) has an odd number of irreducible factors of even degree, a quadratic residue if f(x) has no irreducible factor of even degree or has an even number of them. For, if dl} . . . , 5i are the values of A for the various irreducible factors of f(x), then Asa2$i. . .6t-(raod p), where a is an integer. Hence it suffices to consider an irreducible congruence /(x) = 0 (mod p). Let v be its degree and i a root. In "-1 l~l k i y=IL IL (xp -xp) Z-l Jfc=0 replace x by the v roots; we get two distinct values if v is even, one if v is odd. In the respective cases, $/2==A (mod p) is irreducible or reducible. R. Dedekind82 noted that, if P(x) is a prune function of degree/ modulo p? a prime, a congruence F(a;)==0 (mod p, P) is equivalent to the congruence jp(a) = 0 (mod TT), where TT is a prime ideal factor of p of norm p', and a is a root of P(a) = 0 (mod T). A. E. Pellet83 denoted by /(z)=0 the equation of degree <t>(k) having as its roots the primitive kth roots of unity, and by/i(i/) = 0 the equation derived by setting y = x+l/x. If p is a prune not dividing k, f(x) is congruent modulo p to a product of <t>(k)/v irreducible factors whose degree v is the least integer for which pv — 1 is divisible by k. If fi(y)s=Q (mod p) has an integral root a, /(re) is divisible modulo p by x2— 2az-hl. Either the latter has two real roots and f(x) and fi(y) have all their roots real and p — 1 is divisible by k, or it is irreducible and /(#) is a product of quadratic factors modulo p and the roots of fi(y) are all real and p-fl is divisible by k. If k divides neither p+1 nor p — 1, fi(y) is a product of factors of equal degree modulo p. [Cf. Sylvester,29 etc., Ch. XVI.] Let & be a divisor 5^2 of p+1- Let X be an odd number divisible by no prime not a factor of &, and relatively prime to (p + 1) /k. Then z2X — 2a:rx-f 1 is irreducible modulo p [Serret,74 No. 355]. Also, if b is not divisible by p is irreducible modulo p; replacing x2 by y, we obtain a function of degree X irreducible modulo p. If k is a divisor ?^2 of p — 1 and if X is odd, prime to (p — !)/& and divisible by no prune not a factor of k, F decomposes modulo p into two irreducible functions of degree X. The function /(x2) is either irreducible or the product of two irreducible factors of degree v. In the respective cases, the product A of the squares of 81Comptes Rendus Paris, 86, 1878, 1071-2. 82Abhand. K. Gesell. Wiaa. Gottingen, 23, 1878, p. 25. Dirichlet-Dedekind, Zahlentheorie, ed. 4, 1894, 571-2. 83Comptes Rendus Paris, 90, 1880, 1339-41.