Primitive normal polynomials with a prescribed coefficient |
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Authors: | Shuqin Fan Xiaozhe Wang |
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Institution: | aDepartment of Applied Mathematics, Information Engineering University, Zhengzhou 450002, PR China |
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Abstract: | In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let n15 be a positive integer and q a prime power. We prove that for any aFq and any 1m<n, there exists a primitive normal polynomial f(x)=xn−σ1xn−1++(−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. |
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Keywords: | Finite field Primitive polynomial Character sums Normal basis Galois rings Hansen– Mullen conjecture |
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