Normal bases and irreducible polynomials

TL;DR

This paper explores the relationship between normal bases and irreducible polynomials over finite fields, providing new insights and a unified approach to existing results on N-polynomials and their properties.

Contribution

It offers an alternative proof and unified framework for understanding when irreducible polynomials are N-polynomials based on trace and degree conditions.

Findings

All previous results can be derived from the main theorem.

Established a counting equivalence between N-polynomials and irreducible polynomials with nonzero trace.

Provided a new perspective simplifying the classification of N-polynomials.

Abstract

Let $\mathbb{F}_q$ denote the finite field of $q$ elements and $\mathbb{F}_{q^n}$ the degree $n$ extension of $\mathbb{F}_q$. A normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F} _q$ is a basis of the form $\{\alpha,\alpha^q,\dots,\alpha^{q^{n-1}}\}$. An irreducible polynomial in $\mathbb{F} _q[x]$ is called an $N$-polynomial if its roots are linearly independent over $\mathbb{F} _q$. Let $p$ be the characteristic of $\mathbb{F} _q$. Pelis et al. showed that every monic irreducible polynomial with degree $n$ and nonzero trace is an $N$-polynomial provided that $n$ is either a power of $p$ or a prime different from $p$ and $q$ is a primitive root modulo $n$. Chang et al. proved that the converse is also true. By comparing the number of $N$-polynomials with that of irreducible polynomials with nonzero traces, we present an alternative treatment to this problem and show that all the results mentioned above can be easily deduced from our main theorem.