An approach to normal polynomials through symmetrization and symmetric reduction

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.