Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

$p$-adic formal series and primitive polynomials over finite fields
HTML articles powered by AMS MathViewer

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
Similar Articles
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