Failures of integral Springer's Theorem

TL;DR

This paper investigates the phenomenon where certain elements in number fields are not initially represented by a quadratic form but become so over specific extensions, revealing it occurs infinitely often with finiteness results for fixed forms.

Contribution

It demonstrates that failures of integral representation over a number field can be remedied over odd degree totally real extensions, and establishes finiteness results for fixed quadratic forms.

Findings

Failures occur infinitely often over number fields.

Representation can be achieved over odd degree totally real extensions.

Finiteness results are established for fixed quadratic forms.

Abstract

We discuss the phenomenon where an element in a number field is not integrally represented by a given positive definite quadratic form, but becomes integrally represented by this form over a totally real extension of odd degree. We prove that this phenomenon happens infinitely often, and, conversely, establish finiteness results about the situation when the quadratic form is fixed.