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We study the zeros (mod p) of the polynomial /P ( X) = -A (BA/k)(XP--' 1) forp an odd prime, where BA denotes the k th Bernoulli number and the summation extends over 1 < k < p 2. We establish a reciprocity law which relates the congruence jP ( r) 0 (mod p) to a congruence fp ( n) 0 O (mod r) for r a prime less than p and n E Z. The polynomialfp(x) is the irreducible polynomial over Q of the number Tr9() ~, where D is a primitive p2 th root of unity and L c Q(D) is the extension of degree p over Q. These congruences are closely related to the prime divisors of the indices I(a) = (62: Z[a]), where 6 is the integral closure in L and a E 6 is of degree p over Q. We establish congruences (mod p) involving the numbers I(a) and show that their prime divisors r 0 p are closely related to the congruence rP 1 (mod p2). 0. Introduction. If the Bernoulli numbers Bk, k = 0,1,..., are given by the expansion t 00 tk t= E Bkk! then one defines, for p an odd prime, the polynomial p2B I3P(x) = E k (Xp-lk _ 1). k =1 k Note that the coefficients of this polynomial are p-integral (Kummer). In this paper we prove the equivalence of the congruence /3p(r) 0 (mod p), 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 D be a primitive p2th root of unity (p an odd prime). Then Gal(Q(D)/Q) contains a unique subgroup H of order p 1. Let L be the corresponding fixed field. We define (0.1) Hp =Tr?QM?. If one identifies Gal(Q(D)/Q) with (Z/p2 Z) X in the usual way, then one finds that H a{P (mod p2) 1 < a < p 1}. Hence (0.2) Hp=

A reciprocity law for polynomials with Bernoulli coefficients