A reciprocity law for polynomials with Bernoulli coefficients

Abstract:

We study the zeros $\pmod p$ of the polynomial ${\beta _p}(X) = {\Sigma _k}({B_k}/k)({X^{p - 1 - k}} - 1)$ for $p$ an odd prime, where ${B_k}$ denotes the $k$th Bernoulli number and the summation extends over $1 \leqslant k \leqslant p - 2$. We establish a reciprocity law which relates the congruence ${\beta _p}(r) \equiv 0\;\pmod p$ to a congruence ${f_p}(n) \equiv 0 \pmod r$ for $r$ a prime less than $p$ and $n \in {\mathbf {Z}}$. The polynomial ${f_p}(x)$ is the irreducible polynomial over ${\mathbf {Q}}$ of the number $\operatorname {Tr}_L^{{\mathbf {Q}}(\zeta )}\zeta$, where $\zeta$ is a primitive ${p^2}$ th root of unity and $L \subset {\mathbf {Q}}(\zeta )$ is the extension of degree $p$ over ${\mathbf {Q}}$. These congruences are closely related to the prime divisors of the indices $I(\alpha ) = (\mathcal {O}:{\mathbf {Z}}[\alpha ])$, where $\mathcal {O}$ is the integral closure in $L$ and $\alpha \in \mathcal {O}$ is of degree $p$ over ${\mathbf {Q}}$. We establish congruences $\pmod p$ involving the numbers $I(\alpha )$ and show that their prime divisors $r \ne p$ are closely related to the congruence ${r^{p - 1}} \equiv 1 \pmod {p^2}$.