Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Jacobi sums and new families of irreducible polynomials of Gaussian periods

Author: F. Thaine
Journal: Math. Comp. 70 (2001), 1617-1640
MSC (2000): Primary 11R18, 11R21, 11T22
Published electronically: May 11, 2001
MathSciNet review: 1836923
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


Let $m> 2$, $\zeta _m $ an $m$-th primitive root of 1, $q\equiv 1$ mod $2m$ a prime number, $s=s_{q}$ a primitive root modulo $q$ and $f=f_{q}=(q-1)/m$. We study the Jacobi sums $J_{a,b}=-\sum _{k=2}^{q-1}\zeta _m ^{\, a\, \text{ind}_{s}(k)+b\, \text{ind}_{s}(1-k)}$, $0\leq a, b\leq m-1$, where $\text{ind}_{s}(k)$ is the least nonnegative integer such that $s^{\, \text{ind}_{s}(k)}\equiv k$ mod $q$. We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families $P_{q}(x)$, $q\in \mathcal{P}$, of irreducible polynomials of Gaussian periods, $\eta _{i}=\sum _{j=0}^{f-1}\zeta _q^{s^{i+mj}}$, of degree $m$, where $\mathcal{P}$ is a suitable set of primes $\equiv 1$ mod $2m$. We exhibit examples of such families for several small values of $m$, and give a MAPLE program to construct more of them.

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

  • 1. B. Berndt, R. Evans and K. Williams, Gauss and Jacobi sums, John Wiley & Sons Inc., New York-Toronto, 1998. MR 99d:11092
  • 2. L.E. Dickson, Cyclotomy, higher congruences and Waring's problem, Amer. J. Math. 57 (1935), 391-424.
  • 3. H. Edwards, Fermat's Last Theorem, a Genetic Introduction to Algebraic Number Theory, Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin-Heidelberg, 1977. MR 83b:12001
  • 4. 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
  • 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:11067a
  • 7. R. Schoof and L. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), 543-556. MR 89h:11067b
  • 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, Trans. Amer. Math. Soc. 351 (1999), 4769-4790. MR 2000c:11181
  • 12. F. Thaine, Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers, Math. Comp. 69 (2000), 1653-1666. MR 2001a:11179
  • 13. L. C. Washington, Introduction to Cyclotomic Fields, Second Edition, Graduate Texts in Mathematics, Springer-Verlag, New York, 1996. MR 97h:11130
  • 14. A. Weil, Jacobi sums as ``Grössencharaktere'', Trans. Amer. Math. Soc. 73 (1952), 487-495. MR 14d:452d

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11R18, 11R21, 11T22

Retrieve articles in all journals with MSC (2000): 11R18, 11R21, 11T22

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): September 15, 1998
Received by editor(s) in revised form: January 19, 2000
Published electronically: May 11, 2001
Additional Notes: This work was supported in part by grants from NSERC and FCAR
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society