Further Results on the Morgan–Mullen Conjecture.

Let F q be the finite field of characteristic p with q elements and F q n 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 F q n / F 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 ∤ n . [ABSTRACT FROM AUTHOR]