Let $F:=\mathbb{Q}(\sqrt{1+\sqrt{3}})$ and $\alpha=\sqrt{1+\sqrt{3}}$. I have to find the minimal polynomial of $\alpha$ over $\mathbb{Q}$ and over $\mathbb{Q}(\sqrt{3})$. I also need to show that $\mathbb{Q}(\sqrt{3})$ is contained in $F$.
My attempt, the minimal polynomial over $\mathbb{Q}$ is $x^4-2x^2-2$, which is irreducible by Eisenstein's criterion. How to find the minimal polynomial over $\mathbb{Q}(\sqrt{3})$?
And for the second one $\sqrt{3}=\alpha^2-1$, that's how we can show the containment. Right?