The Coefficient-Choosing Game

Abstract

Let $D$ be an integral domain. Two players, Nora and Wanda, alternately choose coefficients from $D$ for a polynomial of degree $d$. When they are done, if the polynomial has a root in the field of fractions of $D$, then Wanda wins. If not, then Nora wins. We determine, for many $D$, who wins this game.