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-adic formal series and primitive polynomials over finite fields

Authors:	Shuqin Fan Wenbao Han

Institution:	Department of Applied Mathematics, Information Engineering University, Zhengzhou, 450002, People's Republic of China ; Department of Applied Mathematics, Information Engineering University, Zhengzhou, 450002, People's Republic of China

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 -adic number fields and the estimates of character sums over Galois rings. Given we prove, for large enough , 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 with the -th ( coefficient $a_{m}$ fixed in advance except when $m=\frac{n+1}{2}$ if is odd and when $m=\frac{n}{2}, \frac{n}{2}+1$ if is even.

Keywords:	Finite field primitive polynomial character sums over Galois rings $p$-adic formal series

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