Generalised quadratic forms and the u-invariant

TL;DR

This paper extends the concept of the u-invariant to hermitian and generalized quadratic forms over division algebras with involution in characteristic 2, exploring their relationships and specific cases like quaternion algebras.

Contribution

It introduces new u-invariants for hermitian and generalized quadratic forms and analyzes their properties and relationships in characteristic 2 fields.

Findings

Defined u-invariants for hermitian and generalized quadratic forms

Analyzed relationships between these invariants

Studied cases of quaternion division algebras

Abstract

The u-invariant of a field is the supremum of the dimensions of anisotropic quadratic forms over the field. We define corresponding u-invariants for hermitian and generalised quadratic forms over a division algebra with involution in characteristic 2 and investigate the relationships between them. We also investigate these invariants in the case of a quaternion algebra and in particular when this quaternion algebra is the unique quaternion division algebra over a field.