# Witt rings of quadratically presentable fields

## TL;DR

This paper develops a new axiomatic framework for quadratic forms using presentable partial orders, enabling a uniform construction of Witt rings across all field characteristics.

## Contribution

It introduces the concept of quadratically presentable fields and demonstrates their use in constructing Witt rings uniformly for all characteristics.

## Key findings

- Witt rings over fields are isomorphic to those over quadratically presentable fields.
- The framework applies to fields of any characteristic, including characteristic 2.
- A new categorical approach to quadratic forms is established.

## Abstract

This paper introduces a novel approach to the axiomatic theory of quadratic forms. We work internally in a category of certain partially ordered sets, subject to additional conditions which amount to a strong form of local presentability. We call such partial orders presentable. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of quadratically presentable fields, that is fields equipped with a presentable partial order inequationaly compatible with the algebraic operations. As an application, we show that Witt rings of symmetric bilinear forms over fields, of both characteristic 2 and not 2, are isomorphic to Witt rings of suitably built quadratically presentable fields, which therefore provide a uniform construction of Witt rings for all characteristics.