Quintic rings over Dedekind domains and their sextic resolvents

Abstract

Bhargava parametrized quintic rings over $\mathbb{Z}$ by quadruples of $5\times 5$ alternating matrices. We extend the construction to work similarly over any Dedekind domain $R$. No assumptions are needed on the characteristic of $R$. The resolvent consists of a pair of locally free modules $L$, $M$ with two multilinear maps between them; we can view $L$ as $Q/R$, for $Q$ the quintic ring, and $M$ as $S/R$, where $S$ is a sextic resolvent ring. As in Bhargava's treatment, any quintic ring has a resolvent ring, and for a maximal ring, the resolvent is unique. We hope that this work will enable the removal of the condition that the characteristic be different from $2$ in Bhargava-Shankar-Wang's proof of Linnik's conjecture on the asymptotic distribution of discriminants of relative extensions.