Composition of Binary Quadratic Forms over Number Fields

TL;DR

This paper generalizes the classical correspondence between ideal class groups and binary quadratic forms from quadratic number fields to arbitrary base fields with narrow class number one, providing explicit descriptions and conditions.

Contribution

It extends the classical theory to broader number fields and offers explicit descriptions and conditions for quadratic forms in this generalized setting.

Findings

Generalization of the ideal class group and quadratic form correspondence to arbitrary base fields

Explicit description of the correspondence in the generalized setting

Conditions for total positivity in forms with totally negative discriminants

Abstract

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.