A Reciprocity Law for Polynomials

We study the zeros (mod p) of the polynomial ßp(X) = ¿Zk(Bk/k)(Xp~l~k 1) for p an odd prime, where Bk denotes the k\\\ Bernoulli number and the summation extends over 1 < k < p — 2. We establish a reciprocity law which relates the congruence ßp(r) = 0 (mod p) to a congruence f (n) = 0 (mod r) for r a prime less than/7 and n e Z. The polynomial/ (x) is the irreducible polynomial over Q of the number Tr^(il f, where f is a primitive p2 th root of unity and L c Q(f ) is the extension of degree p over Q. These congruences are closely related to the prime divisors of the indices 1(a) = (0 : T\a]), where 0 is the integral closure in L and a e 0 is of degree p over Q. We establish congruences (mod p ) involving the numbers /(a) and show that their prime divisors r =£ p are closely related to the congruence rr~l ■ 1 (mod p2 ). 0. Introduction. If the Bernoulli numbers Bk, k = 0,1,..., are given by the expansion t S tk — 7 = £ 5/c7T> e'-l k=0 kl then one defines, for p an odd prime, the polynomial ß,(x)-Z%-(x>-l-*-i). A = l K Note that the coefficients of this polynomial are/?-integral (Kummer). In this paper we prove the equivalence of the congruence ß (r) = 0 (mod/?), where r is a prime such that r < p, to a polynomial congruence (mod r). In order to construct these polynomial congruences, we introduce a class of cyclic extensions of Q Let i be a primitive p2th root of unity (p an odd prime). Then Gal(Q(f)/Q) contains a unique subgroup H of order p — 1. Let L be the corresponding fixed field. We define (0.1) Hp = Tr«. If one identifies Gal(Q(f)/Q) with (Z/p2Z)x in the usual way, then one finds that H = {a^ímod p2): l<a</?-l). Hence (0.2) hp= i r. 1<a<p-1 Received by the editors July 5, 1983. 1980 Mathematics Subject Classification. Primary 12A35; Secondary 12A50.