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 ${r^{p - 1}} \equiv 1 \pmod {p^2}$.References
-
L. E. Dickson, History of the theory of numbers: Vol. 1, Stechen, New York, 1934.
- W. L. Fouché, Arithmetic properties of Heilbronn sums, J. Number Theory 19 (1984), no. 1, 1–6. MR 751160, DOI 10.1016/0022-314X(84)90088-X K. Hensel, Arithmetische Untersuchungen über die gemeinsamen ausserwesentlichen Discriminantentheiler einer Gattung, J. Reine Angew. Math. 113 (1894), 128-160.
- D. H. Lehmer, On Fermat’s quotient, base two, Math. Comp. 36 (1981), no. 153, 289–290. MR 595064, DOI 10.1090/S0025-5718-1981-0595064-5 D. H. Lehmer and E. Lehmer, Cyclotomy for non-squarefree moduli, Proc. Analytic Number Theory (Philadelphia, 1980), Lecture Notes in Math., vol. 899, Springer-Verlag, Berlin and New York. 1981. pp. 276-300.
- Neal Koblitz, $p$-adic numbers, $p$-adic analysis, and zeta-functions, Graduate Texts in Mathematics, Vol. 58, Springer-Verlag, New York-Heidelberg, 1977. MR 0466081, DOI 10.1007/978-1-4684-0047-2 B. Mazur, Analyse $p$-adique, Bourbaki report, 1972.
- Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, Monografie Matematyczne, Tom 57, PWN—Polish Scientific Publishers, Warsaw, 1974. MR 0347767
- Robert A. Smith, On $n$-dimensional Kloosterman sums, J. Number Theory 11 (1979), no. 3, S. Chowla Anniversary Issue, 324–343. MR 544261, DOI 10.1016/0022-314X(79)90006-4
- André Weil, Basic number theory, 3rd ed., Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag, New York-Berlin, 1974. MR 0427267, DOI 10.1007/978-3-642-61945-8
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