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

DOI:
https://doi.org/10.1090/S0025-5718-99-01142-4

Published electronically:
May 19, 1999

MathSciNet review:
1653998

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Abstract | References | Similar Articles | Additional Information

Abstract: Given an odd prime we show a way to construct large families of polynomials , , where is a set of primes of the form mod and is the irreducible polynomial of the Gaussian periods of degree in . Examples of these families when are worked in detail. We also show, given an integer and a prime mod , how to represent by matrices the Gaussian periods of degree in , and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of .

**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 : 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

Email:
ftha@vax2.concordia.ca

DOI:
https://doi.org/10.1090/S0025-5718-99-01142-4

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