$\mathcal O_K$ is a PID where $K=\mathbb{Q}(\sqrt{-2})$

Question

I want to show that $\mathcal O_K$ is a principal ideal domain where $K=\mathbb{Q}(\sqrt{-2})$. Then use this to show that every prime $p \equiv 1,3$ (mod 8) can be written as $p=x^2+2y^2$ with $x,y \in \mathbb{Z}$.

Answer 1

Because the Minkowski bound of $K=\mathbb{Q}(\sqrt{-2})$ satisfies $B_K\sim 1.8006<2$, the class group is trivial, i.e., $\mathcal{O}_K$ is a PID. Now the norm of any element $z$ in $\mathcal{O}_K$ is given by $N(z)=z\overline{z}=a^2+2b^2$ for $z=a+b\sqrt{-2}$. Then use the answers for this MSE-question:

Proving that if $p^2 = a^2 + 2b^2$ then also $p$ can be written in form $a^2 +2b^2$