Sums of integral squares in complex bi-quadratic fields and in CM fields

Abstract: Let $K$ be a complex bi-quadratic field with ring of integers $\mathcal{O}{K}$. For $K = \mathbb{Q}(\sqrt{-m}$, $\sqrt{n}$), where $ m \equiv 3 \pmod 4 $ and $ n \equiv 1 \pmod 4$, we prove that every algebraic integer can be written as sum of integral squares. Using this, we prove that for any complex bi-quadratic field $K$, every element of $4\mathcal{O}_K$ can be written as sum of five integral squares. In addition, we show that the Pythagoras number of ring of integers of any CM field is at most five. Moreover, we give two classes of complex bi-quadratic fields for which $p(\mathcal{O}{K})= 3$ and $p(4\mathcal{O}{K})=3 $ respectively. Here, $p(\mathcal{O}{K})$ is the Pythagoras number of ring of integers of $K$ and $p(4\mathcal{O}{K})$ is the smallest positive integer $t$ such that every element of $4\mathcal{O}{K}$ can be written as sum of $t$ integral squares.