For the polynomial $t^4-2$ in $\Bbb Q[t]$, the splitting field is given by $\Bbb Q(\alpha, i)$ where $\alpha$ is $2^{1/4}$.
I figured out that the Galois group of this polynomial is the dihedral group $D_4$, whose order is $8$.
And also I found that there are $10$ subgroups of $D_4$, which means that there are $10$ intermediate fields between $\Bbb Q(\alpha, i)$ and $\Bbb Q$ (including themselves).
But, I keep failing to find what those intermediate fields actually are. I found $\Bbb Q$, $\Bbb Q(\alpha, i)$, $\Bbb Q(\alpha)$ ,$\Bbb Q(\alpha i)$, $\Bbb Q(\alpha^2)$, $\Bbb Q(\alpha^2i)$, $\dots$, but need $4$ more. Could anyone show me what the remaining fields are?