Consider $K=\mathbb Q(\sqrt{2},\sqrt{3})$ - I think $K$ is Galois since it's the splitting field of $(x-\sqrt 2)(x+\sqrt 2)(x-\sqrt 3)(x+\sqrt 3)$. I feel like $G(K/F)$ is isomorphic to the Klein 4 group since you can swap $\pm\sqrt{2}$, $\pm\sqrt{3}$ or both (since the roots have no F-relation). But since $K$ is Galois, $|G(K/F)|=[K:F]$ which means $[K:F]=4$, which clashes with my intuition that it should have degree three (the basis units being 1, $\sqrt{2}$, $\sqrt 3$).
What am I not understanding?