On biquadratic fields: when 5 squares are not enough

TL;DR

This paper investigates the Pythagoras number of rings of integers in totally real biquadratic fields, providing new results and supporting evidence for a conjecture relating field properties to this number.

Contribution

It advances the understanding of the Pythagoras number in biquadratic fields, proving the conjecture for fields containing  or , and refines the list of exceptional cases.

Findings

Proved the conjecture for fields containing  or .

Showed all but finitely many fields with  or  satisfy  .

Presented computational evidence supporting the conjecture for fields containing .

Abstract

In this paper we study the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ for the rings of integers in totally real biquadratic fields $K$. We continue the work of Tinkov\'a towards proving the conjecture by Kr\'asensk\'y, Ra\v{s}ka and Sgallov\'a that a biquadratic $K$ satisfies $\mathcal{P}(\mathcal{O}_K)\geq 6$ if and only if it contains neither $\sqrt{2}$ nor $\sqrt{5}$, with only finitely many exceptions. We fully solve two out of three remaining classes of fields by proving that all but finitely many $K$ containing $\sqrt{6}$ or $\sqrt{7}$ satisfy $\mathcal{P}(\mathcal{O}_K)\geq 6$. Furthermore, we present ideas and computations which further support the conjecture also for $K$ containing $\sqrt{3}$. This enables us to refine the conjecture by explicitly listing the exceptional fields.