Pythagoras numbers of orders in biquadratic fields

TL;DR

This paper investigates the Pythagoras number of rings of integers in biquadratic fields, showing that the maximum known bound is attained in many cases while most fields have a lower Pythagoras number of 5.

Contribution

It establishes the exact Pythagoras number for a broad class of biquadratic fields and identifies the typical and extremal values in this setting.

Findings

Maximum Pythagoras number 7 is attained in many fields.

Most fields have Pythagoras number 5.

Lower bounds of 5 and 6 are established for various fields.

Abstract

We examine the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ of the ring of integers $\mathcal{O}_K$ in a totally real biquadratic number field $K$. We show that the known upper bound $7$ is attained in a large and natural infinite family of such fields. In contrast, for almost all fields $\mathbb{Q}(\sqrt5, \sqrt{s})$ we prove $\mathcal{P}(\mathcal{O}_K)=5$. Further we show that $5$ is a lower bound for all but seven fields $K$ and $6$ is a lower bound in an asymptotic sense.