ON THE k-NORMAL ELEMENTS AND POLYNOMIALS OVER FINITE FIELDS

Abstract

An element α ∈ \(\mathbb{F}_{q^{n}}\) is normal over \(\mathbb{F}_{q}\) if the set {α, α , ..., α\(^{q^{n-1}}\)} is a basis of \(\mathbb{F}_{q^{n}}\) over \(\mathbb{F}_{q}\).  The k-normal elements over finite fields are defined and characterized by Huczynska, Mullen, Panario and Thomson (2013).  For 0 ≤ k ≤ n−1, the element α ∈ \(\mathbb{F}_{q^{n}}\) is said to be a k-normal element if \(gcd(x^{n} − 1, \sum_{i=0}^{n-1} α^{q^{i}} x^{n-1-i})\) has degree k.  It is well known that a 0-normal element is a normal element.  So, the k-normal elements are a generalization of normal elements.  By analogy with the case of normal polynomials, a monic irreducible polynomial of degree n is called a k-normal polynomial if its roots are k-normal elements of \(\mathbb{F}_{q^{n}}\) over \(\mathbb{F}_{q}\).  In this paper, a new characterization and construction of k-normal elements and polynomials over finite fields are given.