Pythagoras number of quartic orders containing $\sqrt{2}$

Abstract

Let $K$ be a quartic number field containing $\sqrt{2}$ and let $\mathcal{O}\subseteq K$ be an order such that $\sqrt{2}\in \mathcal{O}$. We prove that the Pythagoras number of $\mathcal{O}$ is at most 5. This confirms a conjecture of Krásenský, Raška and Sgallová. The proof makes use of Beli's theory of bases of norm generators for quadratic lattices over dyadic local fields.

v3: numbering system changed to be consistent with the published version. Chinese version published in Chinese Ann. Math. Ser. A 45 (2024), no. 3, 287--296.; English translation in Chinese Journal of Contemporary Mathematics, 45 (2024) no. 3