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Mathematics of Computation

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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 $\mathbb{Q}(\eta _{0})$.

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. K. Iwasawa, A note on Jacobi sums, Symposia Mathematica 15 (1975), 447-459. MR 52:5620
  • 3. S. Lang, Cyclotomic fields I and II (with an appendix by K. Rubin), Combined Second Edition, Graduate Texts in Mathematics, Springer-Verlag, New York, 1990. MR 91c:11001
  • 4. V. A. Lebesgue, Recherches sur les nombres, J. Math. Pures Appl. 2 (1837), 253-292.
  • 5. E. Lehmer, The quintic character of 2 and 3, Duke Math. J. 18 (1951), 11-18. MR 12:677a
  • 6. E. Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), 535-541. MR 89h:10067a
  • 7. R. Schoof and L. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), 543-556. MR 89h:10067b
  • 8. T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, Markham Publishing Company, Chicago, 1967. MR 36:128
  • 9. H.W. Lloyd Tanner, On the binomial equation $x^{p}-1=0$: quinquisection, Proc. London Math. Soc. 18 (1886/87), 214-234.
  • 10. F. Thaine, Properties that characterize Gaussian periods and cyclotomic numbers, Proc. Amer. Math. Soc. 124 (1996), 35-45. MR 96d:11115
  • 11. F. Thaine, On the coefficients of Jacobi sums in prime cyclotomic fields, Transactions of the American Mathematical Society, to appear.
  • 12. L. C. Washington, Introduction to Cyclotomic Fields, Second Edition, Graduate Texts in Mathematics, Springer-Verlag, New York, 1996. MR 97h:11130

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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

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

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