For which rational values of $a$ is $\mathbb{Q}(\sqrt{\sqrt{5} + a}) / \mathbb{Q}$ a normal field extension?
My main approach to solving this has been to look at when $\mathbb{Q}(\sqrt{\sqrt{5} + a})$ is the splitting field of a polynomial over $\mathbb{Q}$. I've shown that $\sqrt{\sqrt{5} + a}$ is a root of the polynomial $x^4 -2ax^2 + a^2 -5$, however I'm not sure when $\mathbb{Q}(\sqrt{\sqrt{5} + a})$ is a splitting field for this polynomial.