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$p$-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
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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

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

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