$p$-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 Free Access

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 $p$-adic number fields and the estimates of character sums over Galois rings. Given $n$ we prove, for large enough $q$, the Hansen-Mullen conjecture that there exists a primitive polynomial $f(x)=x^{n}-a_{1}x^{n-1}+\cdots +(-1)^{n}a_{n}$ over $F_{q}$ of degree $n$ with the $m$-th ($0<m<n)$ coefficient $a_{m}$ fixed in advance except when $m=\frac {n+1}{2}$ if $n$ is odd and when $m=\frac {n}{2}, \frac {n}{2}+1$ if $n$ is even.

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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

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