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arXiv · 1612.05931

Existence results on k-normal elements over finite fields

Lucas Reis
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Abstract

An element $\alpha \in \mathbb {F}_{q^n}$ is normal over $\mathbb {F}_q$ if $\alpha$ and its conjugates $\alpha, \alpha^q, \cdots \alpha^{q^{n-1}}$ form a basis of $\mathbb {F}_{q^n}$ over $\mathbb {F}_q$. Recently, Huczynska, Mullen, Panario and Thomson (2013) introduce 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 $\mathbb {F}_q$.

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Lucas Reis. 2016-12-18. Existence results on k-normal elements over finite fields. https://arxiv.org/abs/1612.05931

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