Existence results on k-normal elements over finite fields

An element alpha is an element of F-q(n) is normal over Fq if a and its conjugates alpha, alpha(q), ..., alpha(qn-1) form a basis of F-q(n) over F-q. In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of k-normal elements, generalizing the normal elements. In the past few years, many questions concerning the existence and number of k-normal elements with specified properties have been proposed. In this paper, we discuss some of these questions and, in particular, we provide many general results on the existence of k-normal elements with additional properties like being primitive or having large multiplicative order. We also discuss the existence and construction of k-normal elements in finite fields, providing a connection between k-normal elements and the factorization of x(n) - 1 over F-q.