An odd degree descent problem for quasi-subforms of quadratic forms

Abstract

Let $\phi$ and ψ be regular quadratic forms over a field F of characteristic different from 2. We say that ψ is a quasisubform of $\phi$ if there is $a\in F^{*}$ such that $a\psi$ is a subform of $\phi$. Let L/F be an odd degree field extension. Assume that ψ_L is a quasisubform of ${\phi }_L$. A natural question is whether ψ is a quasisubform of $\phi$. We give a positive answer to this question in any of the following cases:

1. The 2-cohomological dimension of F equals 2.

2. $\text{dim\hspace{0.17em}}\phi -\text{dim\hspace{0.17em}}\psi \le 1,$ or $\text{dim\hspace{0.17em}}\phi -\text{dim\hspace{0.17em}}\psi =2$ and $\text{dim\hspace{0.17em}}\phi$ is even.

3. $\text{dim\hspace{0.17em}}\psi =3,$ and $\text{dim\hspace{0.17em}}\phi \le 6.$

4. $\text{dim\hspace{0.17em}}\psi =3,$ and $⟪a,b⟫I^2(F)=0,$ where $⟪a,b⟫$ is the Pfister form associated with ψ (the most difficult case).

Keywords:

• Quadratic form
• Scharlau’s transfer
• Witt ring

2020 Mathematics Subject Classification:

• 11E04
• 11E81