Forms over fields and Witt's lemma

TL;DR

This paper reviews the framework of forms over fields, relates it to classical forms, and proves a version of Witt's lemma demonstrating transitivity of isometry groups on subspaces.

Contribution

It extends the theory of forms over fields and establishes a Witt's lemma variant, connecting classical and modern form theories.

Findings

Unified framework for forms over fields

Proved Witt's lemma for these forms

Showed transitivity of isometry groups on subspaces

Abstract

We give an overview of the general framework of forms of Bak, Tits and Wall, when restricting to vector spaces over fields, and describe its relationship to the classical notions of Hermitian, alternating and quadratic forms. We then prove a version of Witt's lemma in this context, showing in particular that the action of the group of isometries of a space equipped with a form is transitive on isometric subspaces.