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Transactions of the American Mathematical Society

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



A reciprocity law for polynomials with Bernoulli coefficients

Author: Willem Fouché
Journal: Trans. Amer. Math. Soc. 288 (1985), 59-67
MSC: Primary 11R18; Secondary 11R09
MathSciNet review: 773047
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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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Keywords: Bernoulli numbers, Fermat quotients, discriminants, cyclic abelian extensions
Article copyright: © Copyright 1985 American Mathematical Society