2023/09/11 by Connolly, Darien, George, Calvin, Hou, Xiang-dong +2
#05E05 #11T06 #12E20 #12F10 #20C05 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2309.05470
An irreducible polynomial f∈\Bbb Fq[X] of degree n is \em normal over \Bbb Fq if and only if its roots r, rq,…,r^qn-1 satisfy the condition Δn(r, rq,…,r^qn-1)≠ 0, where Δn(X0,…,Xn-1) is the n× n circulant determinant. By finding a suitable \em 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∈\Bbb 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.