I have two questions:

Determine the degree of the extension degree of $\mathbb{Q}(\sqrt{2 + \sqrt{7}})$ over $\Bbb Q$ and the degree of the splitting field of the minimal polynomial of $\sqrt{2 + \sqrt{7}}$ over $\Bbb Q$.

For the first one, that's what I have thought:

$$x =\sqrt{2 + \sqrt{7}} \implies x^2 = 2 + \sqrt7 \implies x^4 - 4x^2 -3 = 0$$

Now $m_{a}(x) = x^4 - 4x^2 - 3$ is irreducible in $\mathbb{Q}$ and it is the minimal polynomial of $\sqrt{2 + \sqrt{7}}$, so the degree of $\mathbb{Q}(x)$ should be $4$. Am I correct?

About the second question: how can I find the degree of the splitting field of $m_{a}(x)$? The only thing I know is that $m_{a}(x)$ has $4$ roots, which are $\pm\sqrt{2 \pm \sqrt{7}}$. What now?


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