= Solution
Put $\alpha=\sqrt[4]7$. The roots of $T^4-7$ are $\alpha,i\alpha,-\alpha,-i\alpha$, so its <splitting field> is
$$
L=\mathbb Q(\alpha,i).
$$
The polynomial is <irreducible polynomial>[irreducible] over $\mathbb Q$ by the Eisenstein criterion at $7$, and $\mathbb Q(\alpha)\subset\mathbb R$ does not contain $i$. The <tower law for field extensions> therefore gives $[L:\mathbb Q]=8$.
Define automorphisms
$$
r(\alpha)=i\alpha,quad r(i)=i,qquad
s(\alpha)=\alpha,quad s(i)=-i.
$$
They satisfy $r^4=s^2=1$ and $srs=r^{-1}$. The eight maps $r^j$ and $r^js$ are distinct, so they exhaust the Galois group. Hence
$$
\operatorname{Gal}(L/\mathbb Q)\cong D_8,
$$
the <dihedral group> of order eight.
The <Galois correspondence> gives all intermediate fields. The full list, with a single generator for each field, is
$$
\begin{array}{c|c|c}
\text{subgroup}&\text{fixed field}&\text{normal over }\mathbb Q\\ \hline
D_8&\mathbb Q=\mathbb Q(0)&\text{yes}\\
\langle r\rangle&\mathbb Q(i)&\text{yes}\\
\langle r^2,s\rangle&\mathbb Q(\sqrt7)&\text{yes}\\
\langle r^2,rs\rangle&\mathbb Q(\sqrt{-7})&\text{yes}\\
\langle r^2\rangle&\mathbb Q(\sqrt7+i)&\text{yes}\\
\langle s\rangle&\mathbb Q(\alpha)&\text{no}\\
\langle r^2s\rangle&\mathbb Q(i\alpha)&\text{no}\\
\langle rs\rangle&\mathbb Q((1+i)\alpha)&\text{no}\\
\langle r^3s\rangle&\mathbb Q((1-i)\alpha)&\text{no}\\
\{1\}&L=\mathbb Q(\alpha+i)&\text{yes}.
\end{array}
$$
For example, $\sqrt7+i$ has stabilizer $\langle r^2\rangle$, and $(1+i)\alpha$ has stabilizer $\langle rs\rangle$; the other entries follow similarly. Also $\alpha+i$ has trivial stabilizer, so its orbit has eight elements and the <primitive element theorem> conclusion $L=\mathbb Q(\alpha+i)$ holds explicitly.
Finally, the <normal subextension criterion> says that normal intermediate fields correspond exactly to normal subgroups. In $D_8$ these are $D_8$, $\langle r\rangle$, the two index-two Klein four-groups, $\langle r^2\rangle$, and $\{1\}$, which gives exactly the “yes” rows above. This is the <subfields of the splitting field of x to the fourth minus seven> lattice.
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