A Criterion for the Normality of Polynomials over Finite Fields Based on Their Coefficients

TL;DR

This paper introduces a polynomial criterion based on coefficients to determine the normality of irreducible polynomials over finite fields, providing explicit formulas for degrees up to 6 and extending results to arbitrary fields with cyclic Galois groups.

Contribution

It defines a universal polynomial condition for polynomial normality over finite fields and computes it explicitly for degrees up to 5, with partial results for degree 6, also extending to arbitrary fields.

Findings

Explicit polynomial $h_n$ computed for $n \\le 5$

A simplified polynomial $h_{p,n}$ depending on characteristic $p$

Results valid for fields with cyclic Galois groups

Abstract

An irreducible polynomial over $\Bbb F_q$ is said to be normal over $\Bbb F_q$ if its roots are linearly independent over $\Bbb F_q$. We show that there is a polynomial $h_n(X_1,\dots,X_n)\in\Bbb Z[X_1,\dots,X_n]$, independent of $q$, such that if an irreducible polynomial $f=X^n+a_1X^{n-1}+\cdots+a_n\in\Bbb F_q[X]$ is such that $h_n(a_1,\dots,a_n)\ne 0$, then $f$ is normal over $\Bbb F_q$. The polynomial $h_n(X_1,\dots,X_n)$ is computed explicitly for $n\le 5$ and partially for $n=6$. When $\text{char}\,\Bbb F_q=p$, we also show that there is a polynomial $h_{p,n}(X_1,\dots,X_n)\in\Bbb F_p[X_1,\dots,X_n]$, depending on $p$, which is simpler than $h_n$ but has the same property. These results remain valid for monic separable irreducible polynomials over an arbitrary field with a cyclic Galois group.