Minimal quadratic forms for the function field of a conic in characteristic $2$

TL;DR

This paper constructs explicit minimal quadratic forms of dimensions 5 and 7 over the function field of a conic in characteristic 2, advancing understanding of quadratic form behavior in this setting.

Contribution

It provides explicit examples of minimal quadratic forms in characteristic 2 and introduces a method leveraging cyclic p-algebras and Witt index bounds.

Findings

Explicit minimal quadratic forms of dimensions 5 and 7 constructed.

Uses properties of cyclic p-algebras to describe quadratic forms.

Establishes bounds on Witt index for certain quadratic forms.

Abstract

In this note, we construct explicit examples of $F_Q$-minimal quadratic forms of dimension $5$ and $7$, where $F_Q$ is the function field of a conic over a field $F$ of characteristic $2$. The construction uses the fact that any set of $n$ cyclic $p$ algebras over a field of characteristic $p$ can be described using only $n+1$ elements of the base field. It also uses a general result that provides an upper bound on the Witt index of an orthogonal sum of two regular anisotropic quadratic forms over a henselian valued field.