We study the polynomial f(x)=xq+1+ax+b over an arbitrary field F of characteristic p, where q is a power of p and ab≠0. The polynomial has arisen recently in several different contexts, including the inverse Galois problem, difference sets, and Müller–Cohen–Matthews polynomials in characteristic 2. We prove f has exactly n rational roots, where n∈{0,1,2,Q+1} and F∩GF(q)=GF(Q). If F is finite then we find the exact number of a,b∈F× such that f has n rational roots, for each n. We also prove many arithmetic properties of f. For example, if F is finite and f has a rational root r, then f has exactly two rational roots if and only if NF/GF(Q)(r−1)≠1. The techniques rely on a detailed analysis of the splitting field and Galois group, together with frequent use of Hilbert's Theorem 90.
