On quadratic forms of height two and a theorem of Wadsworth

Abstract

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