Compute the degree of the extension $$\mathbb{Q}\left(\sqrt{1+i\sqrt3}+\sqrt{1-i\sqrt3}\right)$$ over $\mathbb{Q}$
Note the radicals extend over both the imaginary and real parts. I've looked at splitting fields and theorems about extension fields but I have no idea what to do, like at all here. It makes sense to me for something like $$\mathbb{Q}(\sqrt{3}\mkern1.5mu)$$ or something like that, but I'm lost here. All I know is to find the minimal polynomial. We also have not studied Galois theory yet, I found explanations elsewhere but they referenced Galois so I don't really understand them. Any concrete place to start would be wonderful. Thanks!