A criterion for the normality of polynomials over finite fields based on their coefficients

Abstract

An irreducible polynomial over Fq is said to be normal over Fq if its roots are linearly independent over Fq. We show that there is a polynomial hn(X1,…,Xn)∈Z[X1,…,Xn], independent of q, such that if an irreducible polynomial f=Xn+a1Xn−1+⋯+an∈Fq[X] is such that hn(a1,…,an)≠0, then f is normal over Fq. The polynomial hn(X1,…,Xn) is computed explicitly for n≤5 and partially for n=6. When charFq=p, we also show that there is a polynomial hp,n(X1,…,Xn)∈Fp[X1,…,Xn], depending on p, which is simpler than hn but has the same property. These results remain valid for monic separable irreducible polynomials over an arbitrary field with a cyclic Galois group.