If $E= \mathbb{Q}(\sqrt{2} + \sqrt{7})$ Show that $E/\mathbb{Q}$ is a Galois Extension.

Question

If $E= \mathbb{Q}(\sqrt{2} + \sqrt{7})$ Show that $E/\mathbb{Q}$ is a Galois Extension.

I know that $E/Q$ must be a normal and separable extension.

I can also show that

(1) E is a splitting field of a separable polynomial with coefficients in F.

(2) Every irreducible polynomial in F[x] with at least one root in E splits over E and is separable.

How can I determine which approach will be better, or there is another approach to solve this?. Thanks.

Answer 1

Hint: Notice that $\Bbb{Q}(\sqrt{2}+\sqrt{7})=\Bbb{Q}(\sqrt{2},\sqrt{7})$ which is the splitting field of:

$(x^2-2)(x^2-7)$

So, I guess in this case the way that you see this is by finding a polynomial of which your extension field is the splitting field by using the adjoined roots.