𝑝-adic formal series and primitive polynomials over finite fields
Shuqin Fan, Wenbao Han · Proceedings of the American Mathematical Society · 2003
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 p -adic number fields and the estimates of character sums over Galois rings. Given n n we prove, for large enough q q , the Hansen-Mullen conjecture that there exists a primitive polynomial f ( x ) = x n − a 1 x n − 1 + ⋯ + ( − 1 ) n a n f(x)=x^{n}-a_{1}x^{n-1}+\cdots +(-1)^{n}a_{n} over F q F_{q} of degree n n with the m m -th ( 0 > m > n ) 0>m>n) coefficient a m a_{m} fixed in advance except when m = n + 1 2 m=\frac {n+1}{2} if n n is odd and when