ON THE k-NORMAL ELEMENTS AND POLYNOMIALS OVER FINITE FIELDS

An element alpha is an element of F-qn is normal over F-q if the set {alpha, alpha(q), ..., alpha(qn-1)} is a basis of F-qn over 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 alpha is an element of F-qn is said to be a k-normal element if gcd(x(n) -1, Sigma(n-1)(i=0) alpha(qi) 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 F-qn over F-q. In this paper, a new characterization and construction of k-normal elements and polynomials over finite fields are given.