Find the Galois group of $\mathbb{Q}(\sqrt{4+\sqrt{7}})/\mathbb{Q}$

Question

Let $E = \mathbb{Q}(\sqrt{4+\sqrt{7}})$. Prove that $E$ is a normal extension of $\mathbb{Q}$ and find the Galois group $Gal(E/\mathbb{Q})$.

My attempt:

Let $\alpha = \sqrt{4+\sqrt{7}}$. $\alpha$ is a root of the polynomial $(x^2 - 4)^2 - 7$.

1) How can I show $f(x) = (x^2 - 4)^2 - 7 = x^4 - 8x^2 + 9$ is irreducible over $\mathbb{Q}$ because I can't apply Eisenstein condition?

If $f(x)$ is irreducible, $[E:\mathbb{Q}] = 4 = |Gal(E/\mathbb{Q})|$ and I have shown that all roots of $f(x)$, say $\alpha_{1} = \sqrt{4+\sqrt{7}}, \alpha_{2} = -\sqrt{4+\sqrt{7}}, \alpha_{3} = \sqrt{4-\sqrt{7}}, \alpha_{4} = -\sqrt{4-\sqrt{7}}$ are in $E$ and hence $E$ is a normal extension of $\mathbb{Q}$. Let $Gal(E/\mathbb{Q}) = \{ e,\sigma,\tau,\sigma\tau \}$

2) How can I find the order of $\sigma,\tau$ to find $Gal(E/\mathbb{Q})$? Thanks.

Answer 1

Note that $E$ is a field extension of $\mathbb{Q}(\sqrt{7})$ and hence we can use the tower rule:

$4\geq [E:\mathbb{Q}]=[E:\mathbb{Q}(\sqrt{7})][\mathbb{Q}(\sqrt{7}):\mathbb{Q}]=[E:\mathbb{Q}(\sqrt{7})]\times 2$

The element $\alpha$ does not belong to $\mathbb{Q}(\sqrt{7})$. In order to see this try to write $\alpha=a+b\sqrt{7}$ when $a,b\in\mathbb{Q}$ and you will get that $\sqrt{7}$ is rational which is of course a contradiction. Hence $[E:\mathbb{Q}(\sqrt{7})]$ is at least $2$, so that gives us $4\geq[E:\mathbb{Q}]\geq 4$ which of course implies $[E:\mathbb{Q}]=4$.

Ok, now you know that the extension is Galois of degree $4$. Hence $Gal(E/\mathbb{Q})$ is either $\mathbb{Z_4}$ or $\mathbb{Z_2}\times\mathbb{Z_2}$. However, if we denote $\beta=\sqrt{4-\sqrt{7}}$ then the roots of $f$ are $\alpha,-\alpha,\beta,-\beta$. Now, since the extension $E/\mathbb{Q}$ is simple and generated by $\alpha$ we know that for any $c\in E$ which is a root of the minimal polynomial of $\alpha$ there exists an element in $Gal(E/\mathbb{Q})$ which sends $\alpha\to c$. Hence there is an element in $Gal(E/\mathbb{Q})$ which sends $\alpha\to -\alpha$. Since $\beta=\frac{3}{\alpha}$ it is easy to see that this automorphism sends $\beta\to -\beta$, and hence it defines the permutation $(\alpha,-\alpha)(\beta,-\beta)$ on the roots of $f$. Also, there is an element in $Gal(E/\mathbb{Q})$ which sends $\alpha\to\beta$. Again, it is easy to check that such an automorphism must send $\beta\to\alpha$, and hence it defines the permutaiton $(\alpha,\beta)(-\alpha,-\beta)$. So $Gal(E/\mathbb{Q})$ contains at least two elements of order $2$ and this implies that it must be isomorphic to $\mathbb{Z_2}\times\mathbb{Z_2}$.

Answer 2

HINTS.

1. If it were reducible, then it would decompose into two quadratics, a linear and a cubic, or four linear factors. To have a linear factor, it would have to have a root over $\mathbb{Q}$. Does it? This only leaves the quadratic case. Assume it is a product $(x^2+ax+b)(x^2+cx+d)$ and see if you can arrive at a contradiction with the coefficients.

2. Here recall what I like to think of as 'conjugation for quadratics': $\overline{a+b \sqrt{d}}= a-b\sqrt{d}$. Can you see 'two' of these 'conjugations' (one 'outside' and one 'inside') for the defining element for your field? This should make it easier to see what the order of the elements must be by actually writing down the maps explicitly. The Galois group should then be apparent.