Primitive normal pairs with prescribed norm and trace

Abstract

Let q be a prime power and n be a positive integer. In this article, we consider the problem of the existence of a primitive normal element α in Fqn of a prescribed primitive norm a and non-zero trace b over Fq, such that f(α) is also a primitive normal element in Fqn, where f(x)∈Fqn[x] is an arbitrary polynomial with minor restrictions. For this existence problem, we obtain a sufficient condition and show that for n≥5 there exist at least (q−1)ϕ(q−1) primitive normal elements α such that f(α) is also a primitive normal element in all but finitely many fields Fqn. In particular, for q=7k;k≥1, deg(f(x))=2 and n≥8, we find that there are only 5 fields in which the existence of such elements is not guaranteed.