An approach to normal polynomials through symmetrization and symmetric reduction

TL;DR

This paper investigates conditions for irreducible polynomials over finite fields to be normal, using symmetrization of circulant determinants, and extends the analysis to specific polynomial forms and Galois extensions.

Contribution

It introduces a symmetrization approach to determine normality of polynomials, providing explicit conditions for certain polynomial forms and extending the theory to Galois extensions.

Findings

Symmetrization yields a sufficient condition for polynomial normality.

Explicit conditions for normality of degree 6 and 7 polynomials of form X^n+X^{n-1}+a.

Extension of normal polynomial analysis to finite Galois extensions via character theory.

Abstract

An irreducible polynomial $f\in\Bbb F_q[X]$ of degree $n$ is {\em normal} over $\Bbb F_q$ if and only if its roots $r, r^q,\dots,r^{q^{n-1}}$ satisfy the condition $\Delta_n(r, r^q,\dots,r^{q^{n-1}})\ne 0$, where $\Delta_n(X_0,\dots,X_{n-1})$ is the $n\times n$ circulant determinant. By finding a suitable {\em symmetrization} of $\Delta_n$ (A multiple of $\Delta_n$ which is symmetric in $X_0,\dots,X_{n-1}$), we obtain a condition on the coefficients of $f$ that is sufficient for $f$ to be normal. This approach works well for $n\le 5$ but encounters computational difficulties when $n\ge 6$. In the present paper, we consider irreducible polynomials of the form $f=X^n+X^{n-1}+a\in\Bbb F_q[X]$. For $n=6$ and $7$, by an indirect method, we are able to find simple conditions on $a$ that are sufficient for $f$ to be normal. In a more general context, we also explore the normal polynomials of a finite Galois extension through the irreducible characters of the Galois group.