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?