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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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A reciprocity law for polynomials with Bernoulli coefficients
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by Willem Fouché PDF
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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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 288 (1985), 59-67
  • MSC: Primary 11R18; Secondary 11R09
  • DOI: https://doi.org/10.1090/S0002-9947-1985-0773047-X
  • MathSciNet review: 773047