Universal Quadratic Forms and Indecomposables over Biquadratic 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.

14 pages, comments are welcome