Are these simple field extensions normal? Learning for an upcoming exam on basic algebra I came across the following problem:

Test for $\alpha = \sqrt{2+\sqrt{2}}$ and $\alpha = \sqrt{1+\sqrt{3}}$ whether the field extension $\mathbb{Q}(\alpha)/\mathbb{Q} $ is normal.

I feel, that I should be able to solve it, but I am running against a wall here. Can you explain to me, how one would generally approach this kind of problem?

Best Regards
Take the first one, $\alpha =\sqrt {2+\sqrt 2}$.  Let's find it's minimal polynomial.  We remark that $$\alpha^2=2+\sqrt 2\implies \left(\alpha^2-2\right)^2=2$$  so we have $$\alpha^4-4\alpha^2+2$$  That is effectively quadratic and we can solve to get the roots $$\pm \sqrt {2\pm \sqrt 2}$$.

Letting $\beta=\sqrt {2-\sqrt 2}$ we easily see that $$\alpha \beta = \sqrt 2\implies \beta =\frac {\alpha^2-2}{\alpha}\implies \beta\in \mathbb Q(\alpha)$$ 

The other instance should follow along similar lines.
Since $\mathbb{Q}$ has characteristic zero, you only have to check that $\alpha$'s minimal polynomial has no multiple roots.