Properties that characterize Gaussian periods and cyclotomic numbers
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- by F. Thaine PDF
- Proc. Amer. Math. Soc. 124 (1996), 35-45 Request permission
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 $\Bbb Q(\zeta _q)$. Write $\eta _0\eta _i=\sum _{j=0}^{e-1} a_{i,j}\eta _j$ with $a_{i,j}\in \Bbb Z$. 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 $\Bbb Q$, of the $\eta _i$ or on the irreducibility, over $\Bbb Q$, of the characteristic polynomial of the matrix $[a_{i,j}]_{0\leq i,j\leq e-1}$.References
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L. E. Dickson, Cyclotomy, higher congruences and Waring’s problem, Amer. J. Math. 57 (1935), 391–424.
- Thomas Storer, Cyclotomy and difference sets, Lectures in Advanced Mathematics, No. 2, Markham Publishing Co., Chicago, Ill., 1967. MR 0217033 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).
- Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
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
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 35-45
- MSC (1991): Primary 11R18; Secondary 11T22
- DOI: https://doi.org/10.1090/S0002-9939-96-03108-5
- MathSciNet review: 1301532