关于系数选择博弈
On the coefficient-choosing game
摘要：
Nora和Wanda是两个参与人，他们从某个固定的单位交换环$R$中选择一个次数$d$多项式的系数。如果多项式的根在$R$分式环中，则万达被宣布为获胜者，否则诺拉被宣布为获胜者。我们将Gasarch，Washington和Zbarsky给出的这些博弈的理论推广到所有有限循环环，并确定了可能的结果。一个家庭的例子也构造使用离散赋值环的变种，这些作者提出的游戏。我们的技术有导致我们的对抗性的方法来构建合理的多项式的任何规定的程度（等于$3$或大于$8$）没有根的最大阿贝尔扩展$\mathbb{Q}$。
英文摘要：
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 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 $\mathbb{Q}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
