An approach to normal polynomials through symmetrization and symmetric reduction

An irreducible polynomial f∈Fq[X] of degree n is normal over Fq if and only if its roots r,rq,…,rqn−1 satisfy the condition Δn(r,rq,…,rqn−1)≠0, where Δn(X0,…,Xn−1) is the n×n circulant determinant. By finding a suitable symmetrization of Δn (A multiple of Δn which is symmetric in X0,…,Xn−1), we obtain a condition on the coefficients of f that is sufficient for f to be normal. This approach works well for n≤5 but encounters computational difficulties when n≥6. In the present paper, we consider irreducible polynomials of the form f=Xn+Xn−1+a∈Fq[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.