Existence results on kk-normal elements over finite fields

  • Lucas Reis

    Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
Existence results on $k$-normal elements over finite fields cover
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An element αFqn\alpha \in \mathbb{F}_{q^n} is normal over Fq\mathbb{F}_q if α\alpha and its conjugates α,αq,,αqn1\alpha, \alpha^q, \dots, \alpha^{q^{n-1}} form a basis of Fqn\mathbb{F}_{q^n} over Fq\mathbb{F}_q. In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of kk-normal elements, generalizing the normal elements. In the past few years, many questions concerning the existence and number of kk-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 kk-normal elements with additional properties like being primitive or having large multiplicative order. We also discuss the existence and construction of kk-normal elements in finite fields, providing a connection between kk-normal elements and the factorization of xn1x^n-1 over Fq\mathbb{F}_q.

Cite this article

Lucas Reis, Existence results on kk-normal elements over finite fields. Rev. Mat. Iberoam. 35 (2019), no. 3, pp. 805–822

DOI 10.4171/RMI/1070