Let $L:=\mathbb{Q}(\sqrt{3}+\sqrt[3]{5})$. How to determine $[L:\mathbb{Q}]$ and $\mathrm{Gal}(L/K)$?
I thought about showing $\mathbb{Q}(\sqrt{3}+\sqrt[3]{5})=\mathbb{Q}(\sqrt{3},\sqrt[3]{5})$ first.
It's $\sqrt{3}+\sqrt[3]{5} \in \mathbb{Q}(\sqrt{3},\sqrt[3]{5}) \Rightarrow \mathbb{Q}(\sqrt{3}+\sqrt[3]{5})\subset\mathbb{Q}(\sqrt{3},\sqrt[3]{5})$
The other direction
$(\sqrt{3}+\sqrt[3]{5})^2=3+2\sqrt{3}\sqrt[3]{5}+(\sqrt[3]{5})^2$
Here I don't see what to do next. What has to be done now?