# Group and round quadratic forms

## TL;DR

This paper provides new characterisations of group and round quadratic forms, explores their properties over various field extensions, and clarifies conditions for forms to become anisotropic group forms that are not round.

## Contribution

It introduces elementary characterisations of group and round quadratic forms and establishes new results on their behavior over Laurent series fields and extensions.

## Key findings

- New characterisations of group and round quadratic forms
- Results on 'going-up' properties over Laurent series fields
- Conditions for forms to become anisotropic non-round group forms

## Abstract

We offer some elementary characterisations of group and round quadratic forms. These characterisations are applied to establish new (and recover existing) characterisations of Pfister forms. We establish "going-up" results for group and anisotropic round forms with respect to iterated Laurent series fields, which contrast with the established results with respect to rational function field extensions. For forms of two-power dimension, we determine when there exists a field extension over which the form becomes an anisotropic group form that is not round.