# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## A reciprocity law for polynomials with Bernoulli coefficientsHTML articles powered by AMS MathViewer

by Willem Fouché
Trans. Amer. Math. Soc. 288 (1985), 59-67 Request permission

## 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 ${r^{p - 1}} \equiv 1 \pmod {p^2}$.
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