Is Q(4th root(2)):Q(sqrt(2)) a Galois extension?

Question

I'm having difficulty with this little exercise. I know it is a Galois extension but I don't know how to prove it formally. Could someone help me? Thanks.

Answer 1

$\sqrt[4]{2}$ is a root of $x^2 - \sqrt{2} = p(x)\in \mathbb Q(\sqrt{2})$.

$\mathbb Q(\sqrt[4]{2})/ \mathbb Q(\sqrt{2})$ is a normal extension as the roots of $p$ are $\pm\sqrt[4]{2}$. As $\mathbb Q(\sqrt[4]{2})/ \mathbb Q(\sqrt{2})$ is also separable, it is Galois.