On the coefficient-choosing game

Abstract

Nora and Wanda are two players who choose coefficients of a degree-$d$ polynomial from some fixed unital commutative ring $R$. Wanda is declared the winner if the polynomial has a root in the ring of fractions of $R$ and Nora is declared the winner otherwise. We extend the theory of these games given by Gasarch, Washington, and Zbarsky (2018) to all finite cyclic rings and determine the possible outcomes. A family of examples is also constructed using discrete valuation rings for a variant of the game proposed by these authors. Our techniques there lead us to an adversarial approach to constructing rational polynomials of any prescribed degree (equal to 3 or greater than 8) with no roots in the maximal abelian extension of $\mathbb{Q}$.