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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Families of irreducible polynomials
of Gaussian periods and matrices
of cyclotomic numbers


Author: F. Thaine
Journal: Math. Comp. 69 (2000), 1653-1666
MSC (1991): Primary 11R18, 11R21, 11T22
Published electronically: May 19, 1999
MathSciNet review: 1653998
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Abstract | References | Similar Articles | Additional Information

Abstract: Given an odd prime $p$ we show a way to construct large families of polynomials $P_{q}(x)\in \mathbb{Q}[x]$, $q\in \mathcal{C}$, where $\mathcal{C}$ is a set of primes of the form $q\equiv 1$ mod $p$ and $P_{q}(x)$ is the irreducible polynomial of the Gaussian periods of degree $p$ in $\mathbb{Q}(\zeta _{q})$. Examples of these families when $p=7$ are worked in detail. We also show, given an integer $n\geq 2$ and a prime $q\equiv 1$ mod $2n$, how to represent by matrices the Gaussian periods $\eta _{0},\dots ,\eta _{n-1}$ of degree $n$ in $\mathbb{Q}(\zeta _{q})$, and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of $\mathbb{Q}(\eta _{0})$.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01142-4
PII: S 0025-5718(99)01142-4
Received by editor(s): May 19, 1998
Received by editor(s) in revised form: October 15, 1998
Published electronically: May 19, 1999
Additional Notes: This work was supported in part by grants from NSERC and FCAR
Article copyright: © Copyright 2000 American Mathematical Society