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

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?

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.