I'd like to compute the intermediate fields between $\mathbb{Q}$ and a splitting field for $f=X^4-10X^2+4$.
Here's what I've done:
The roots of $f$ are $\pm \sqrt{5\pm \sqrt{21}}$. Multiplying them we see that a splitting field for $f$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt{5+\sqrt{21}})$ which has degree 4 over $\mathbb{Q}$.
Hence the order of the Galois group is 4. Its elements are determined by the action on $\sqrt{5+\sqrt{21}}$, which can be sent to either of the four roots of $f$. It's easily checked that the three non-identity elements have order $2$, hence the Galois group is isomorphic to $C_2\times C_2$.
By the Galois correspondence, there are three intermediate fields. One of them is obviously $\mathbb{Q}(\sqrt{21})$, but how do I get the other ones? I tried to manually determine the fixed fields for all three subgroups, but the calculations got too nasty for me.