Universal Quadratic Forms and Indecomposables over Biquadratic Fields

TL;DR

This paper investigates indecomposable algebraic integers and universal quadratic forms over biquadratic fields, providing conditions, estimates, and bounds that verify Kitaoka's conjecture in specific cases.

Contribution

It offers new criteria for indecomposability in biquadratic fields, develops an algorithmic approach for escalation, and proves bounds on variables of universal forms, confirming Kitaoka's conjecture in two fields.

Findings

Sufficient conditions for indecomposability in biquadratic fields.

Algorithmic estimates for escalation method.

Lower bounds on variables of universal quadratic forms in specific fields.

Abstract

The aim of this article is to study (additively) indecomposable algebraic integers $\mathcal O_K$ of biquadratic number fields $K$ and universal totally positive quadratic forms with coefficients in $\mathcal O_K$. There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field $K$. Furthermore, estimates are proven which enable algorithmization of the method of escalation over $K$. These are used to prove, over two particular biquadratic number fields $\mathbb{Q}(\sqrt{2}, \sqrt{3})$ and $\mathbb{Q}(\sqrt{6}, \sqrt{19})$, a lower bound on the number of variables of a universal quadratic forms, verifying Kitaoka's conjecture.