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

An irreducible polynomial over ${\mathbb{F}}_q$ is said to be normal over ${\mathbb{F}}_q$ if its roots are linearly independent over ${\mathbb{F}}_q$. We show that there is a polynomial $h_n(X_1,\dots ,X_n)\in \mathbb{Z}[X_1,\dots ,X_n]$, independent of $q$, such that if an irreducible polynomial $f=X^n+a_1{X}^{n-1}+\cdots +a_n\in {\mathbb{F}}_q[X]$ is such that $h_n(a_1,\dots ,a_n)\mathrm{\ne }0$, then $f$ is normal over ${\mathbb{F}}_q$. The polynomial $h_n(X_1,\dots ,X_n)$ is computed explicitly for $n\le 5$ and partially for $n=6$. When $\text{char}{\mathbb{F}}_q=p$, we also show that there is a polynomial $h_{p,n}(X_1,\dots ,X_n)\in {\mathbb{F}}_p[X_1,\dots ,X_n]$, 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.