Resumen de Existence results on k-normal elements over finite fields

Lucas Reis

• An element α∈Fqn is normal over Fq if α and its conjugates α,αq,…,αqn−1 form a basis of Fqn over Fq. 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 xn−1 over Fq