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 Kr\'{ a}sensk\'{ y}, Ra\v{ s}ka and Sgallov\'a. The proof makes use of Beli's theory of bases of norm generators for quadratic lattices over dyadic local fields.
Categories: math.NT