A new criterion on k-normal elements over finite fields

TL;DR

This paper introduces a novel criterion for identifying k-normal elements in finite fields using idempotents, expanding the theoretical tools available for finite field extension analysis.

Contribution

It presents a new criterion for k-normal elements based on idempotents, complementing existing methods and criteria in finite field theory.

Findings

New criterion for k-normal elements using idempotents

Examples demonstrating the criterion's application

Extension of criteria previously known for normal elements

Abstract

The notion of normal elements for finite fields extension has been generalized as k-normal elements by Huczynska et al. [3]. The number of k-normal elements for a fixed finite field extension has been calculated and estimated [3], and several methods to construct k-normal elements have been presented [1,3]. Several criteria on k-normal element have been given [1,2]. In this paper we present a new criterion on k-normal elements by using idempotents and show some examples. Such criterion has been given for usual normal element before [6].