Further results on the Morgan-Mullen conjecture

2018/11/01 by Garefalakis, Theodoulos, Kapetanakis, Giorgos · 1 citation
#11T24 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1811.00896

Abstract

Let \mathbbF_q be the finite field of characteristic p with q elements and \mathbbF_qⁿ its extension of degree n. The conjecture of Morgan and Mullen asserts the existence of primitive and completely normal elements (PCN elements) for the extension \mathbbF_qⁿ/\mathbbF_q for any q and n. It is known that the conjecture holds for n ≤ q. In this work we prove the conjecture for a larger range of exponents. In particular, we give sharper bounds for the number of completely normal elements and use them to prove asymptotic and effective existence results for q≤ n≤ O(q^ε), where ε=2 for the asymptotic results and ε=1.25 for the effective ones. For n even we need to assume that q-1\nmid n.

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