Lifting problem for universal quadratic forms over totally real cubic number fields

TL;DR

This paper proves that the only totally real cubic number field admitting a universal quadratic form over its ring of integers is , and no such biquadratic fields exist, solving a specific lifting problem in number theory.

Contribution

It establishes the uniqueness of  as the only totally real cubic field with a universal quadratic form over its integers.

Findings

 is the only such totally real cubic field.

No biquadratic fields admit such universal quadratic forms.

The lifting problem has a unique solution in this context.

Abstract

Lifting problem for universal quadratic forms asks for totally real number fields $K$ that admit a positive definite quadratic form with coefficients in $\mathbb{Z}$ that is universal over the ring of integers of $K$. In this paper, we show that $K=\mathbb{Q}(\zeta_7+\zeta_7^{-1})$ is the only such totally real cubic field. Moreover, we show that there is no such biquadratic field.