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

An irreducible polynomial over Fq is said to be normal over F-q if its roots are linearly independent over F-q. We show that there is a polynomial h(n)(X-1, . . . , X-n) is an element of Z[X-1, . . . , X-n], independent of q, such that if an irreducible polynomial f = X-n + a(1)X(n-1) + center dot center dot center dot + a(n) is an element of F-q[X] is such that h(n)(a(1), . . . , a(n)) not equal 0, then f is normal over F-q. The polynomial h(n)(X-1, . . . , X-n) is computed explicitly for n <= 5 and partially for n = 6. When char F-q = p, we also show that there is a polynomial h(p,n)(X-1, . . . , X-n) is an element of F-p[X-1, . . . , X-n], depending on p, which is simpler than h(n) but has the same property. These results remain valid for monic separable irreducible polynomials over an arbitrary field with a cyclic Galois group. (c) 2023 Elsevier Inc. All rights reserved.