This question is an exercise in Dummit and Foote:
1)I want to find the Galois group of $f(x)=x^4+8x^2+8x+4$.
2)Which subfields of the splitting field of $x^4+8x^2+8x+4$ are Galois over $\Bbb Q$?
3) For the subfields which are Galois over $\Bbb Q$, find the polynomial $f(x) \in \Bbb Q[x]$ for which they are the splitting field over $\Bbb Q$.
I have calculated that $f(x)$ is irreducible moreover, the resolvent cubic $h(x)=x^3-16x^2+48x+64$ is irreducible. Again the discriminant $315392$ is not a square so the Galois group has to $S_4$. So part 1) is done.
Now, the subfields of the Galois group of $x^4+8x^2+8x+4$ which are Galois correspond to the fixed field of a normal subgroup of $S_4$ which are $K_4$ and $A_4$. So in some sense part 2) is solved. If I want to answer this question as a field $Fix(K_4)=\Bbb Q(a_1,\cdots,a_n)$ and $Fix(A_4)=\Bbb Q(b_1,\cdots,b_r)$ then what will be $a_i$ and $b_j$.
I am not getting any clue for part 3). Please help.