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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Properties that characterize Gaussian periods
and cyclotomic numbers


Author: F. Thaine
Journal: Proc. Amer. Math. Soc. 124 (1996), 35-45
MSC (1991): Primary 11R18; Secondary 11T22
MathSciNet review: 1301532
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $q=ef+1$ be a prime number, $\zeta _q$ a $q$-th primitive root of 1 and $\eta _0,\dots ,\eta _{e-1}$ the periods of degree $e$ of $\mathbb{Q}(\zeta _q)$. Write $\eta _0\eta _i=\sum _{j=0}^{e-1} a_{i,j}\eta _j$ with $a_{i,j}\in \mathbb{Z}$. Several characterizations of the numbers $\eta _i$ and $a_{i,j}$ (or, equivalently, of the cyclotomic numbers $(i,j)$ of order $e$) are given in terms of systems of equations they satisfy and a condition on the linear independence, over $\mathbb{Q}$, of the $\eta _i$ or on the irreducibility, over $\mathbb{Q}$, of the characteristic polynomial of the matrix $[a_{i,j}]_{0\leq i,j\leq e-1}$.


References [Enhancements On Off] (What's this?)

  • 1. L. E. Dickson, Cyclotomy, higher congruences and Waring's problem, Amer. J. Math. 57 (1935), 391--424.
  • 2. Thomas Storer, Cyclotomy and difference sets, Lectures in Advanced Mathematics, No. 2, Markham Publishing Co., Chicago, Ill., 1967. MR 0217033 (36 #128)
  • 3. F. Thaine, On the $p$-part of the ideal class group of $\mathbb{Q}(\zeta _p+\zeta _p^{-1})$ and Vandiver's Conjecture, Michigan Math. J. (to appear).
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Additional Information

F. Thaine
Affiliation: address Department of Mathematics and Statistics - CICMA, Concordia University, 1455, de Maisonneuve Blvd. W., Montreal, Quebec, Canada H3G 1M8
Email: ftha@vax2.concordia.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03108-5
PII: S 0002-9939(96)03108-5
Received by editor(s): May 2, 1994
Received by editor(s) in revised form: August 1, 1994
Additional Notes: This work was supported in part by grants from NSERC and FCAR.
Communicated by: William Adams
Article copyright: © Copyright 1996 American Mathematical Society