An element is normal over if and its conjugates form a basis of over . In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of -normal elements, generalizing the normal elements. In the past few years, many questions concerning the existence and number of -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 -normal elements with additional properties like being primitive or having large multiplicative order. We also discuss the existence and construction of -normal elements in finite fields, providing a connection between -normal elements and the factorization of over .
Cite this article
Lucas Reis, Existence results on -normal elements over finite fields. Rev. Mat. Iberoam. 35 (2019), no. 3, pp. 805–822DOI 10.4171/RMI/1070