-adic formal series and primitive polynomials over finite fields

Authors:
Shuqin Fan and Wenbao Han

Journal:
Proc. Amer. Math. Soc. **132** (2004), 15-31

MSC (2000):
Primary 11T55, 11F85, 11L40, 11L07

DOI:
https://doi.org/10.1090/S0002-9939-03-07040-0

Published electronically:
May 8, 2003

MathSciNet review:
2021244

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over -adic number fields and the estimates of character sums over Galois rings. Given we prove, for large enough , the Hansen-Mullen conjecture that there exists a primitive polynomial over of degree with the -th ( coefficient fixed in advance except when if is odd and when if is even.

**1.**S.D.Cohen, Primitive elements and polynomials with arbitrary traces, Discrete Math, vol. 83, no. 1, pp. 1-7, 1990. MR**91h:11143****2.**S.D.Cohen, Primitive elements and polynomials: existence results, Lect. Notes in Pure and Applied Math, vol. 141, edited by G.L.Mullen and P.J.Shiue, Dekker, New York, pp. 43-55, 1993. MR**93k:11113****3.**H.Davenport, Bases for finite fields, J.London Math. Soc, vol. 43, pp. 21-39, 1968. MR**37:2729****4.**B. Dwork, G.Gerotto and F.J.Sullivan, An Introduction to G-Functions, Annals of Mathematics Studies, Number 133, Princeton University Press, 1994. MR**96c:12009****5.**W-B.Han, The coefficients of primitive polynomials over finite fields, Math. of Comp, vol. 65, no. 213, pp. 331-340, Jan. 1996. MR**96d:11128****6.**W-B.Han, On two exponential sums and their applications, Finite Fields and Their Applications, 3, pp. 115-130, 1997.**7.**W-B.Han, On Cohen's Problem, Chinacrypt'96, Academic Press(China), pp. 231-235, 1996 (in Chinese).**8.**W-B.Han, The distribution of the coefficients of primitive polynomials over finite fields, Proceeding of CCNT'99, Prog.in Comp. Sci. and Applied. Logic, vol. 20, Birkhäuser Verlag, Basel/Switzerland, pp. 43-57, 2001.**9.**T.Hansen and G.L.Mullen, Primitive polynomials over finite fields, Math. of Comp, vol. 59, no. 200, pp. 639-643, Supplement: S47-S50, Oct. 1992. MR**93a:11101****10.**T.Helleseth, P.V.Kumar, O.Moreno and A.G.Shanbhag, Improved estimates via exponential sums for the minimum distance of -linear trace codes, IEEE. Trans. Inform. Theory, vol. 42, no.4, pp. 1212-1216, July 1996. MR**97m:94028****11.**D.Jungnickel and S.A.Vanstone, On primitive polynomials over finite fields, J. of Algebra, vol. 124, pp. 337-353, 1989. MR**90k:11164****12.**N. Koblitz, -adic number, -adic analysis and zeta functions, GTM58, Springer-Verlag, 1984. MR**86c:11086****13.**P.V.Kumar, T.Helleseth and A.R.Calderbank, An upper bound for Weil exponential sums over Galois rings and applications, IEEE. Trans. Inform. Theory, vol. 41, no.2, pp. 456-468, Mar. 1995. MR**96c:11140****14.**H.W.Lenstra and R.J.Schoof, Primitive normal bases for finite fields, Math. of Comp, vol. 48, no. 177, pp. 217-232, 1987. MR**88c:11076****15.**W.-C.W.Li, Character sums over -adic fields, J. Number Theory, 74, pp. 181-229, 1999. MR**2000b:11100****16.**R.Lidl, H.Niedereiter, Finite Fields, Addison-Wesley, London, 1983. MR**86c:11106****17.**O.Moreno, On the existence of a primitive quadratic trace over , J. of Combin. Theory Ser. A, vol. 51, pp. 104-110, 1989. MR**90b:11133**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11T55,
11F85,
11L40,
11L07

Retrieve articles in all journals with MSC (2000): 11T55, 11F85, 11L40, 11L07

Additional Information

**Shuqin Fan**

Affiliation:
Department of Applied Mathematics, Information Engineering University, Zhengzhou, 450002, People’s Republic of China

Email:
sq.fan@263.net

**Wenbao Han**

Affiliation:
Department of Applied Mathematics, Information Engineering University, Zhengzhou, 450002, People’s Republic of China

Email:
wb.han@netease.com

DOI:
https://doi.org/10.1090/S0002-9939-03-07040-0

Keywords:
Finite field,
primitive polynomial,
character sums over Galois rings,
$p$-adic formal series

Received by editor(s):
March 13, 2002

Received by editor(s) in revised form:
August 24, 2002

Published electronically:
May 8, 2003

Additional Notes:
This work was supported by NSF of China with contract No. 19971096 and No. 90104035

Communicated by:
Wen-Ching Winnie Li

Article copyright:
© Copyright 2003
American Mathematical Society