Quintic rings over Dedekind domains and their sextic resolvents

Bhargava parametrized quintic rings over Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}$$\end{document} by quadruples of 5×5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\times 5$$\end{document} 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.