element in field, not a square

Question

I am doing a specific exercise where the quaternion group is realised as a Galois group of some field extension. It goes like this: let $K = \mathbb Q(\sqrt 2,\sqrt 3)$ and $\alpha = (2 +\sqrt 2)(3 +\sqrt 6)$. I want to show that $\alpha$ is not a square in $K$. I tried to solve equations explicitly to derive some contradiction but I didn't succeed. Any suggestions? Thanks in advance.