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$p$-ADIC FORMAL SERIES AND COHEN'S PROBLEM

Published online by Cambridge University Press:  15 January 2004

FAN SHUQIN  and
HAN WENBAO
FAN SHUQIN
Affiliation:
Department of Applied Mathematics, Institute of Information Engineering, Information Engineering University, Zhengzhou, 450002, PRC
HAN WENBAO
Affiliation:
Department of Applied Mathematics, Institute of Information Engineering, Information Engineering University, Zhengzhou, 450002, PRC
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Abstract

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With the help of some $p$-adic formal series over $p$-adic number fields and the estimates of character sums over Galois rings, we prove that there is a constant $C(n)$ such that there exists a primitive polynomial $f(x)\,{=}\,x^{n}-a_{1}x^{n-1}+\cdots +(-1)^{n}a_{n}$ of degree $n$ over $F_{q}$ with the first $m=\lfloor\frac{n-1}{2}\rfloor$ coefficients $a_{1},\ldots ,a_{m}$ prescribed in advance if $q\,{>}\,C(n)$.

Keywords

11T5511F8511L40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 46 , Issue 1 , January 2004 , pp. 47 - 61
Copyright
2004 Glasgow Mathematical Journal Trust

Footnotes

This work was supported by NSF of China with contract No. 19971096 and No. 90104035.