Similarity of quadratic and symmetric bilinear forms in characteristic 2

TL;DR

This paper investigates the descent property of quadratic and symmetric bilinear forms in characteristic 2, extending known results to more general forms and algebraic extensions, and proves Scharlau's norm principle in this setting.

Contribution

It extends the descent property for isometry and similarity of quadratic and symmetric bilinear forms to characteristic 2, including arbitrary separable algebraic extensions.

Findings

Odd degree extensions have the descent property for similarity.

Descent property holds for arbitrary separable algebraic extensions in characteristic 2.

Scharlau's norm principle is valid for quadratic and bilinear forms in characteristic 2.

Abstract

We say that a field extension $L/F$ has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over $F$ that become isometric (resp. similar) over $L$ are already isometric (resp. similar) over $F$. The famous Artin-Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic $2$. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic $2$, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau's norm principle for arbitrary quadratic or bilinear forms in characteristic $2$.