An element α ∈ F q n is normal over F q if α and its conjugates α, α q , · · · α q n−1 form a basis of F q n over 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 F q .

The celebrated Primitive Normal Basis Theorem states that for any n ≥ 2 and any finite field F q , there exists an element α ∈ F q n that is simultaneously primitive and normal over F q . In this paper, we prove some variations of this result, completing the proof of a conjecture proposed by Anderson and Mullen (2014). Our results also imply the existence of elements of F q n with multiplicative order (q n − 1)/2 and prescribed trace over F q .

An element α ∈ F q n is normal if B = {α, α q , . . . , α q n−1 } forms a basis of F q n as a vector space over F q ; in this case, B is a normal basis of F q n over F q . The notion of k-normal elements was introduced in Huczynska et al (2013). Using the same notation as before, α is k-normal if B spans a co-dimension k subspace of F q n . It can be shown that 1-normal elements always exist in F q n , and Huczynska et al (2013) show that elements that are simultaneously primitive and 1-normal exist for q ≥ 3 and for large enough n when gcd(n, q) = 1 (we note that primitive 1-normals cannot exist when n = 2). In this paper, we complete this theorem and show that primitive, 1-normal elements of F q n over F q exist for all prime powers q and all integers n ≥ 3, thus solving Problem 6.3 from Huczynska, et al (2013).

Let Fq be the finite field with q elements and f, g ∈ Fq[x] be polynomials of degree at least one. This paper deals with the asymptotic growth of certain arithmetic functions associated to the factorization of the iterated polynomials f (g (n) (x)) over Fq, such as the largest degree of an irreducible factor and the number of irreducible factors. In particular, we provide significant improvements on the results of D. Gómez-Pérez, A. Ostafe and I. Shparlinski (2014).Mathematics Subject Classification (2010): Primary 12E05; Secondary 37P05.