Further results on the Morgan-Mullen conjecture

Official URL: https://dx.doi.org/10.1007/s10623-019-00643-8

Let Fq be the finite field of characteristic p with q elements and Fqn 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 Fqn/Fq 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(qϵ) , 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 ∤ n.