# abstract algebra – Compute the degree of the extension \$mathbb{Q}(sqrt(1+isqrt3)+sqrt(1-isqrt3))\$

Compute the degree of the extension $$mathbb{Q}left(sqrt{1+isqrt3}+sqrt{1-isqrt3}right)$$ 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 $$mathbb{Q}(sqrt(3))$$ 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!