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Primitive normal polynomials with a prescribed coefficient
Authors:Shuqin Fan  Xiaozhe Wang
Institution:aDepartment of Applied Mathematics, Information Engineering University, Zhengzhou 450002, PR China
Abstract:In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let ngreater-or-equal, slanted15 be a positive integer and q a prime power. We prove that for any aset membership, variantFq and any 1less-than-or-equals, slantm<n, there exists a primitive normal polynomial f(x)=xnσ1xn−1+cdots, three dots, centered+(−1)n−1σn−1x+(−1)nσn such that σm=a, with the only exceptions σ1≠0. The theory can be extended to polynomials of smaller degree too.
Keywords:Finite field  Primitive polynomial  Character sums  Normal basis  Galois rings  Hansen–  Mullen conjecture
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