Euclidean Quadratic Forms and ADC Forms I

TL;DR

This paper introduces and studies Euclidean quadratic forms and ADC forms over integral domains, generalizing classical results and initiating their classification over specific algebraic structures.

Contribution

It defines Euclidean and ADC forms over integral domains and proves that Euclidean forms are ADC forms, starting their classification over discrete valuation rings and Hasse domains.

Findings

Euclidean forms are proven to be ADC forms.

Initiation of classification of these forms over specific domains.

Extension of classical results to broader algebraic contexts.

Abstract

Motivated by classical results of Aubry, Davenport and Cassels, we define the notion of a Euclidean quadratic form over a normed integral domain and an ADC form over an integral domain. The aforementioned classical results generalize to: Euclidean forms are ADC forms. We then initiate the study and classification of these two classes of quadratic forms, especially over discrete valuation rings and Hasse domains.