Similarity of quadratic forms and isomorphism of their function fields

Abstract:

This paper considers the question: Given anisotropic quadratic forms $Q$ and $Q’$ over a field $K$ (char $K \ne 2$), if their function fields are isomorphic must $Q$ and $Q’$ be similar? It is proved that the answer is yes if $Q$ is a Pfister form or the pure part of a Pfister form, or a $4$-dimensional form. The argument for Pfister forms and their pure parts does not generalize because these are the only anisotropic forms which attain maximal Witt index over their function fields. To handle the $4$-dimensional case the following theorem is proved: If $Q$ and $Q’$ are two $4$-dimensional forms over $K$ with the same determinant $d$, then $Q$ and $Q’$ are similar over $K$ iff they are similar over $K[\sqrt d ]$. The example of Pfister neighbors suggests that quadratic forms arguments are unlikely to settle the original question for other kinds of forms.