## $p$-adic formal series and primitive polynomials over finite fields

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- by Shuqin Fan and Wenbao Han PDF
- Proc. Amer. Math. Soc.
**132**(2004), 15-31 Request permission

## 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.## References

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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
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2021244