A note on quadratic forms

TL;DR

This paper investigates quadratic forms over field extensions, showing that certain bilinear maps are genuine quadratic forms if and only if the extension is formally unramified, simplifying the axioms in specific cases.

Contribution

It establishes a criterion linking quadratic forms over extensions to formal unramifiedness, revealing that some axioms are unnecessary over finite and number fields.

Findings

All such maps are quadratic forms over $L$ if and only if $L/K$ is formally unramified.

Over finite and number fields, one axiom in the standard quadratic form definition is redundant.

Abstract

For a field extension $L/K$ we consider maps that are quadratic over $L$ but whose polarisation is only bilinear over $K$. Our main result is that all such are automatically quadratic forms over $L$ in the usual sense if and only if $L/K$ is formally unramified. In particular, this shows that over finite and number fields, one of the axioms in the standard definition of quadratic forms is superfluous.