On the number of k-normal elements over finite fields

In this article we give an explicit formula for the number of k -normal elements, hence answering Problem 6.1. of Huczynska et al. (Existence and properties of k -normal elements over finite fields, Finite Fields Appl 2013; 24: 170-183). Furthermore, for some cases we provide formulas that require the solutions of some linear Diophantine equations. Our results depend on the explicit factorization of cyclotomic polynomials.