Generators and irreducible polynomials over finite fields

Author:
Daqing Wan

Journal:
Math. Comp. **66** (1997), 1195-1212

MSC (1991):
Primary 11T24, 11T55

DOI:
https://doi.org/10.1090/S0025-5718-97-00835-1

MathSciNet review:
1401947

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Weil's character sum estimate is used to study the problem of constructing generators for the multiplicative group of a finite field. An application to the distribution of irreducible polynomials is given, which confirms an asymptotic version of a conjecture of Hansen-Mullen.

**[BDS]**E. Bach, J. Driscoll and J. Shallit,*Factor refinement*, J. Algorithm**15**(1993), 199-222. MR**94m:11148****[Ch]**F. R. Chung,*Diameters and eigenvalues*, J. Amer. Math. Soc.**2**(1989), 187-196. MR**89k:05070****[Co]**S. D. Cohen,*Primitive elements and polynomials with arbitrary trace*, Discr. Math.**83**(1990), 1-7. MR**91h:11143****[EH]**G. W. Effinger and D. R. Hayes,*Additive Number Theory of Polynomials over a Finite Field*, Oxford Science Publications, 1991. MR**92k:11103****[Ha1]**W. B. Han,*Some Applications of Character Sums in Finite Fields and Coding Theory*, Ph.D. Dissertation, Sichuan University, 1994.**[Ha2]**W. B. Han,*The coefficients of primitive polynomials over finite fields*, Math. Comp.**65**(213) (1996), 331-340. MR**96d:11128****[HM]**T. Hansen and G. L. Mullen,*Primitive polynomials over finite fields*, Math. Comp.**59**(1992), 639-643. MR**93a:11101****[Ka]**N. M. Katz,*An estimate for character sums*, J. Amer. Math. Soc.**2**(1989), 197-200. MR**90b:11081****[Le1]**H. W. Lenstra Jr.,*Finding isomorphisms between finite fields*, Math. Comp.**56**(1991), 329-347. MR**91d:11151****[Le2]**H. W. Lenstra Jr.,*Multiplicative groups generated by linear expressions*, unpublished notes.**[LS]**H. W. Lenstra Jr. and R. Schoof,*Primitive normal bases for finite fields*, Math. Comp.**48**(1987), 217-231. MR**88c:11076****[Li]**W. C. Li,*Character sums and abelian Ramanujan graphs*, J. Number Theory**41**(1992), 199-217. MR**93h:11092****[Sh]**V. Shoup,*Searching for primitive roots in finite fields*, Math. Comp.**58**(1992), 369-380. MR**92e:11140****[Shp]**I. E. Shparlinski,*Computational and Algorithmic Problems in Finite Fields*, Kluwer Academic Publishers, 1992. MR**94j:11122****[Wa]**D. Wan,*A -adic lifting lemma and its application to permutation polynomials*, Finite Fields, Coding Theory and Advances in Communications and Computing (G. L. Mullen and P. J. S. Shiue, eds.), Marcel Dekker, 1992, pp. 209-216. MR**93m:11129****[We]**A. Weil,*Basic Number Theory*, third edition, Springer-Verlag, 1974. MR**55:302**

Retrieve articles in *Mathematics of Computation*
with MSC (1991):
11T24,
11T55

Retrieve articles in all journals with MSC (1991): 11T24, 11T55

Additional Information

**Daqing Wan**

Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

Email:
wan@math.psu.edu

DOI:
https://doi.org/10.1090/S0025-5718-97-00835-1

Received by editor(s):
December 8, 1995

Received by editor(s) in revised form:
May 8, 1996

Additional Notes:
This research was partially supported by NSF

Article copyright:
© Copyright 1997
American Mathematical Society