Dimensions of anisotropic indefinite quadratic forms II --- The lost proofs

TL;DR

This paper constructs examples of fields with specific invariants related to quadratic forms, including infinite Hasse numbers and prescribed u-invariants, advancing understanding of anisotropic indefinite quadratic forms.

Contribution

It provides new constructions of fields with particular properties of quadratic forms, including finite u-invariants and specific Hasse numbers, with detailed analysis of their orderings.

Findings

Constructed fields with infinite Hasse number and prescribed finite u-invariant.

Established fields where quadratic forms of certain dimensions are Pfister neighbors.

Demonstrated relationships between field invariants and properties of quadratic forms.

Abstract

Let F be a field of characteristic different from 2. The u-invariant and the Hasse number of a field F are classical and important field invariants pertaining to quadratic forms. These invariants measure the suprema of dimensions of anisotropic forms over F that satisfy certain additional properties. We construct various examples of fields with infinite Hasse number and prescribed finite values of u that satisfy additional properties pertaining to the space of orderings of the field. We also construct to each positive integer n a real field F such such that the Hasse number is 2^{n+1} and such that each quadratic form over F of dimension 1+2^n is a Pfister neighbor. These results were announced (without proof) in the article "Dimensions of anisotropic indefinite quadratic forms II" by the present author.