Galois closures of quartic subfields of rational function fields

Abstract

Let k be a finite field with q elements. Let f(x)∈k[x] be a monic quartic polynomial. Then k(x)/k(f(x)) is a field extension of degree 4. If the extension is separable, then the Galois group of the Galois closure is isomorphic to a transitive subgroup of the symmetric group on 4 letters. We determine the number of f(x)ʼs having a given subgroup as Galois group.