Find the Galois group $Gal(\mathbb{Q}[\sqrt{2}]/\mathbb{Q})$ and determine all intermediate subfields explicitly.

Question

Let $K$ be the splitting field of $(x^2-2x-1)(x^2-2x-7)$. Find the Galois group $Gal(K/\mathbb{Q})$ and determine all intermediate subfields explicitly.

I have that $(x^2-2x-1)(x^2-2x-7) = (x-1-\sqrt{2})(x-1+\sqrt{2})(x-1-2\sqrt{2})(x-1+2\sqrt{2})$

Thus $K = \mathbb{Q}[\sqrt{2}]$.

I do not have much practice finding Galois groups, so could somebody outline explicitly how to find the Galois group and the intermediate subgroups?

Answer 1

You did a very good job determining the splitting field: $K=\mathbf{Q}(\sqrt{2})$. This is just a very simple example as you will see.

The extension $K/\mathbf{Q}$ is Galois and has degree two, so the Galois group is $\mathbf{Z}/2\mathbf{Z}$ (the only group, up to isomorphism, of order 2), consisting of the identity automorphism and the one interchanging $\sqrt{2}$ and $-\sqrt{2}$.

There are no non-trivial intermediate fields.