A classical approach to relative quadratic extensions

TL;DR

This paper develops a classical, explicit theory of relative quadratic extensions of number fields, proving a reciprocity law and studying related L-functions, with applications to quadratic equations and class numbers.

Contribution

It introduces a classical approach to relative quadratic extensions, including a reciprocity law and properties of associated L-functions, extending known results to general number fields.

Findings

Proved a reciprocity law relating solutions of quadratic equations to ideal classes.

Defined numbers H(Δ,K) with properties similar to Hurwitz class numbers.

Provided an elementary proof of properties of the Hilbert symbol on higher unit groups.

Abstract

We show that we can develop from scratch and using only classical language a theory of relative quadratic extensions of a given number field $K$ which is as explicit and easy as for the well-known case that $K$ is the field of rational numbers. As an application we prove a reciprocity law which expresses the number of solutions of a given quadratic equation modulo an integral ideal $\mathfrak{a}$ of $K$ in terms of $\mathfrak{a}$ modulo the discriminant of the equation. We study various $L$-functions associated to relative quadratic extensions. In particular, we define, for totally negative algebraic integers $\Delta$ of a totally real number field $K$ which are squares modulo~$4$, numbers $H(\Delta,K)$, which share important properties of classical Hurwitz class numbers. In an appendix we give a quick elementary proof of certain deeper properties of the Hilbert symbol on higher unit groups of dyadic local number fields.