An approach to normal polynomials through symmetrization and symmetric reduction

An irreducible polynomial $f\in F_q\left[X\right]$ of degree n is normal over $F_q$ 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 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 $f=X^n+X^{n-1}+a\in F_q\left[X\right]$. 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.