Quintic polynomials having Galois closure of germs zero with Galois group A_5

摘要

Let f (x) = x(5) + ax(3) + bx(2) + cx, where a, b, and c are elements of the finite field k of size q = p(t), where p > 5 and q equivalent to 1 or 4 (mod 5). We determine those f (x)'s for which the Galois closure of k(x)/k(f (x)) is a rational function field with Galois group A. We then determine the size of the value sets of these polynomials. (C) 2019 Elsevier Inc. All rights reserved.