Non-universality of ternary quadratic forms over fields containing $\sqrt2$

TL;DR

This paper proves that over certain number fields containing 2, no positive definite ternary quadratic form is universal, extending the understanding of quadratic forms in these fields and addressing a conjecture by Kitaoka.

Contribution

It confirms Kitaoka's conjecture for all totally real degree 4 fields and explores properties of ternary quadratic forms over fields with 2, especially where 2 is a square.

Findings

No universal positive definite ternary quadratic forms over degree 4 totally real fields containing 2.

Established properties related to lifting problems for universal quadratic forms.

Provided criteria sets for ternary quadratic forms over these fields.

Abstract

We prove Kitaoka's conjecture for all totally real number fields of degree 4 -- namely, there is no positive definite classical quadratic form in three variables which is universal. To achieve this, we study the fields (often without restricting the degree) where 2 is a square, because in this arguably most difficult case, the recent results connecting Kitaoka's conjecture to sums of integral squares do not apply. We also prove some other properties of ternary quadratic forms over fields containing $\sqrt2$, for example in relation to the lifting problems for universal quadratic forms and for criterion sets.