On quadratic forms of height two and a theorem of Wadsworth

Let $\varphi$ and $\psi$ be anisotropic quadratic forms over a field $F$ of characteristic not $2$. Their function fields $F(% \varphi )$ and $F(\psi )$ are said to be equivalent (over $F$) if $% \varphi \otimes F(\psi )$ and $\psi \otimes F(% \varphi )$ are isotropic. We consider the case where $\dim % \varphi =2^n$ and $% \varphi$ is divisible by an $(n-2)$-fold Pfister form. We determine those forms $\psi$ for which $% \varphi$ becomes isotropic over $F(\psi )$ if $n\leq 3$, and provide partial results for $n\geq 4$. These results imply that if $F(% \varphi )$ and $F(\psi )$ are equivalent and $\dim % \varphi =\dim \psi$, then $% \varphi$ is similar to $\psi$ over $F$. This together with already known results yields that if $% \varphi$ is of height $2$ and degree $1$ or $2$, and if $\dim % \varphi =\dim \psi$, then $F(% \varphi )$ and $F(\psi )$ are equivalent iff $F(% \varphi )$ and $F(\psi )$ are isomorphic over $F$.