adic formal series and primitive polynomials over finite fields
Authors:
Shuqin Fan and Wenbao Han
Journal:
Proc. Amer. Math. Soc. 132 (2004), 1531
MSC (2000):
Primary 11T55, 11F85, 11L40, 11L07
Published electronically:
May 8, 2003
MathSciNet review:
2021244
Fulltext PDF Free Access
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Abstract: In this paper, we investigate the HansenMullen conjecture with the help of some formal series similar to the ArtinHasse exponential series over adic number fields and the estimates of character sums over Galois rings. Given we prove, for large enough , the HansenMullen 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.
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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:
http://dx.doi.org/10.1090/S0002993903070400
PII:
S 00029939(03)070400
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:
WenChing Winnie Li
Article copyright:
© Copyright 2003
American Mathematical Society
