On PAC Extensions and Scaled Trace Forms

TL;DR

This paper extends the isomorphism between non-degenerate quadratic forms and scaled trace forms from Hilbertian fields to broader classes of fields, utilizing PAC extension theory.

Contribution

It generalizes the known result to prosolvable and prime-to extensions of Hilbertian fields using PAC extension theory.

Findings

Non-degenerate quadratic forms over certain fields are isomorphic to scaled trace forms.

Extension to prosolvable and prime-to extensions of Hilbertian fields.

Proofs based on PAC extension theory.

Abstract

Any non-degenerate quadratic form over a Hilbertian field (e.g., a number field) is isomorphic to a scaled trace form. In this work we extend this result to more general fields. In particular, prosolvable and prime-to-p extensions of a Hilbertian field. The proofs are based on the theory of PAC extensions.