On the number of k-normal elements over finite fields

Abstract

In this article we give an explicit formula for the number of k-normal elements, hence answering Problem 6.1. of Huczynska et al. (Existence and properties of k-normal elements over finite fields, Finite Fields Appl 2013; 24: 170-183). Furthermore, for some cases we provide formulas that require the solutions of some linear Diophantine equations. Our results depend on the explicit factorization of cyclotomic polynomials.

Existence and Cardinality of $k$-Normal Elements in Finite Fields

“…This result is independent of the factorization of x m − 1. Moreover, the conditions on m and q required are weaker than the special forms required in [20], and also cannot be derived from the conditions in [18,Theorem 5.5]. Thus, this result extends to cases not previously covered.…”

“…The proof is inspired by the observation in [10] that the additive module structure of F q m in fact gives rise to a group action on all the normal elements. Our bound does not require a specific form for m or q, and therefore extends beyond the formulas provided in [20].…”

“…In certain other cases, it is possible to use information about the factorization of x m − 1 along with Theorem 2 to gain insights into the number of k-normal elements for different values of k. In [20], the authors provide explicit formulas for k-normal elements for degrees m that are either prime powers or numbers of the form 2 v • r, for a prime r = 2, under certain other constraints q and m. Below we state one of their noteworthy results.…”

“…In a recent paper, Saygi et al [20] give formulas (in terms of q and m) for the number of k-normal elements for cases where m is a power of a prime or of the form 2 v • r where r = 2 is a prime and v ≥ 1, using known results on the explicit factorization of cyclotomic polynomials. In particular, their formulae guarantee existence for certain cases.…”

Number of $k$-normal elements over a finite field

“…This formula depends on the factorization of x n − 1 into irreducibles over F q , but since the formula obtained is not "to easy" to handle numerically, Huczynska et al proposed the following problem (see [3,Problem 6.3]): For which values of q, n and k can "nice" explicit formulae (in q and n) be obtained for the number of k-normal elements of F q n over F q ? On this line, in [10] the authors obtained some explicit formulas for certain particular cases of q and n, the results depend on the explicit factorization of cyclotomic polynomials and the solutions of some linear Diophantine equations.…”