Quartic polynomials having Galois closure of genus zero and their value sets

Let v be the number of distinct values of the polynomial f (x) = x(4) + ax(2) bx, where a and b are elements of the finite field k of size q, where q is odd. When b is 0, an exact formula for v can be given. When b is not 0, as a consequence of the Tchebotarev Density Theorem, v = (5/8)q + O(root q). When the Galois closure of k(x)/k(f (x)) is a rational function field, v can be determined exactly. In this work we determine those f(x)'s for which this occurs and give the formula for v. (C) 2017 Elsevier Inc. All rights reserved.