Solution
= Solution
The <Fundamental theorem of Galois theory> says that for a finite <Galois extension> $L/F$ with <group> $G$, the maps $H\mapsto L^H$ and $E\mapsto\operatorname{Gal}(L/E)$ are mutually inverse inclusion-reversing correspondences between subgroups and intermediate fields. They satisfy $[L:L^H]=|H|$ and $[L^H:F]=[G:H]$. An intermediate extension $E/F$ is Galois exactly when its <subgroup> is normal, in which case its <group> is the quotient by that <subgroup>.
Let $a=\sqrt2$, $b=\sqrt3$, $c=i$. The real <field> $\mathbb Q(a,b)$ has degree $4$: if $\sqrt3=u+v\sqrt2$ with $u,v\in\mathbb Q$, squaring forces $uv=0$ and either $u^2=3$ or $v^2=3/2$, both impossible. Adjoining the nonreal $i$ doubles the degree. All three square-root signs can be changed independently, giving
$$
\boxed{G\cong C_2^3,\qquad [L:\mathbb Q]=8.}
$$
Write $\sigma_{uvw}$ for the <automorphism> sending $(a,b,c)$ to $((-1)^ua,(-1)^vb,(-1)^wc)$, where $(u,v,w)\in\mathbb F_2^3$. Every nonzero vector generates an order-two <subgroup>, so there are $7$ such subgroups. Their fixed fields are:
|| Generator
|| Fixed field
| $\sigma_{100}$
| $\mathbb Q(\sqrt3,i)$
| $\sigma_{010}$
| $\mathbb Q(\sqrt2,i)$
| $\sigma_{001}$
| $\mathbb Q(\sqrt2,\sqrt3)$
| $\sigma_{110}$
| $\mathbb Q(\sqrt6,i)$
| $\sigma_{101}$
| $\mathbb Q(\sqrt3,\sqrt{-2})$
| $\sigma_{011}$
| $\mathbb Q(\sqrt2,\sqrt{-3})$
| $\sigma_{111}$
| $\mathbb Q(\sqrt6,\sqrt{-2})$
Each listed <field> has degree $4$ and is fixed by the indicated sign change, which proves equality with the fixed <field>.
There are also $7$ order-four subgroups: they are the kernels of the seven nonzero linear functionals on $\mathbb F_2^3$. For $e=(e_1,e_2,e_3)\ne0$, define $H_e=\{\sigma_{uvw}:e_1u+e_2v+e_3w=0\}$. Then
$$
L^{H_e}=\mathbb Q\left(\sqrt{2^{e_1}3^{e_2}(-1)^{e_3}}\right).
$$
Thus the seven quadratic fields have radicands $2,3,-1,6,-2,-3,-6$. Their square classes are distinct, so all are different. Finally the trivial <subgroup> fixes $L$, and all of $G$ fixes $\mathbb Q$. This accounts for every <subgroup> and intermediate <field>.