Existence results on k-normal elements over finite fields

2016/12/18 by Reis, Lucas · 2 citations
#11T30 and 11T06 #12E20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1612.05931

Abstract

An element α∈ \mathbb F_qⁿ is normal over \mathbb F_q if α and its conjugates α, α^q, ⋯ α^q^n-1 form a basis of \mathbb F_qⁿ 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 xⁿ-1 over \mathbb F_q.

Cited by

• Existence of primitive k-normal elements for critical values over finite fields
• The Minimal Binomial Multiples of Polynomials over Finite Fields

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