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

Abstract

Let K be a quartic number field containing √ 2 and let O ⊆ K be an order such that √ 2 ∈ O. We prove that the Pythagoras number of 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.