Galois group of the simple extension Let $K=Q(\sqrt3,\sqrt5)$. Show that the extension is $K/Q$ is simple and also Galois extension. Determine its Galois group. 
I showed that extension is simple because $K=Q(\sqrt{3}+\sqrt{5})$
But I can not find its Galois group? 

Hint: Notice that  any $\sigma \in \mathrm {Aut} L$ is determined by the images of $\sigma (\sqrt{3})$ and $\sigma (\sqrt 5)$ then the possibilities are $\sigma (\sqrt 3) \in \{-\sqrt 3, \sqrt 3 \}$ and $\sigma (\sqrt 5) \in \{-\sqrt 5, \sqrt 5\}$. 
Can you see how this is related to $\mathbb Z_2 \times \mathbb Z_2$?
