Finding some missing subfields of a splitting field of $x^4-7$ I was looking at the Galois group of the splitting field of $x^4-7$ over $\mathbb{Q}$. I found it to be $\mathbb{Q}(\sqrt[4]{7},i)$, and the Galois group to be the dihedral group of order $8$. Now $D_8=\langle \sigma,\rho\mid \sigma^4=\rho^2=1, \sigma\rho=\rho\sigma^3\rangle$.
I know $D_8$ has $5$ subgroups of index $4$, there should be $5$ subfields of $\mathbb{Q}(\sqrt[4]{7},i)$ of degree $4$ over $\mathbb{Q}$ by the fundamental theorem of Galois theory. 
I found $3$ of them to be $\mathbb{Q}(i\sqrt[4]{7})$, $\mathbb{Q}(\sqrt[4]{7})$, and $\mathbb{Q}(\sqrt{7},i)$. 
I can't figure out what the other $2$ are, but I think they're the fixed fields of the subgroups $\{1,\rho\sigma\}$ and $\{1,\rho\sigma^3\}$, but I can't determine those. Does anyone have any idea what the other two subfields of degree $4$ over $\mathbb{Q}$ are? Many thanks.
 A: Let $a=\sqrt[4]{7}$, $b=i\sqrt[4]{7}$, $c=-\sqrt[4]{7}$, and $d=-i\sqrt[4]{7}$ be the four roots of $x^4-7$.  Then the automorphisms $\rho$ and $\sigma$ permute these roots, with
$$
\rho \,=\, (b\;\;d) \qquad\text{and}\qquad \sigma \,=\, (a\;\;b\;\;c\;\;d)
$$
Observe then that:
$$
\rho\sigma \,=\, (a\;\;d)(b\;\;c), \quad \rho\sigma^2 \,=\, (a\;\;c), \quad \rho\sigma^3\,=\, (a\;\;b)(c\;\;d), \quad\text{and}\quad \sigma^2 \,=\, (a\;\;c)(b\;\;d)
$$
As you suggest, the intermediate fields $\mathbb{Q}(\sqrt[4]{7})$, $\mathbb{Q}(i\sqrt[4]{7})$, and $\mathbb{Q}(i,\sqrt{7})$ correspond to three of the subgroups of order two:
$$
\mathbb{Q}(\sqrt[4]{7}) \leftrightarrow \{1,\rho\},\qquad \mathbb{Q}(i\sqrt[4]{7})\leftrightarrow \{1,\rho\sigma^2\}, \qquad \mathbb{Q}(i,\sqrt{7})\leftrightarrow\{1,\sigma^2\}
$$
The main trick for figuring out the fixed field of $\{1,\rho\sigma\}$ is to consider the symmetrizing function $1+\rho\sigma$.  Because of symmetry, anything in the image of $1+\rho\sigma$ must be fixed under $\rho\sigma$:
$$
\rho\sigma(1+\rho\sigma)(x) \,=\, (\rho\sigma + (\rho\sigma)^2)(x) = (\rho\sigma + 1)(x) = (1+\rho\sigma)(x)
$$
Now, observe that
$$
(1+\rho\sigma)(a) \,=\, a+d = (1-i)\sqrt[4]{7}
$$
and
$$
(1+\rho\sigma)(b) \,=\, b+c = (i-1)\sqrt[4]{7}
$$
Therefore, the fixed field of $\{1,\rho\sigma\}$ is $\mathbb{Q}\bigl((1-i)\sqrt[4]{7}\bigr)$.  Using a similar calculation, we can see that the fixed field of $\{1,\rho\sigma^3\}$ is $\mathbb{Q}\bigl((1+i)\sqrt[4]{7}\bigr)$.
