Originally Posted by

**Jose27** I'm having trouble with this question: Find the splitting field of $\displaystyle f(x)=x^4 - 7$ over $\displaystyle F= \mathbb{Q}$and find all the intermediate fields of the extension.

The splitting field is $\displaystyle \mathbb{Q} (\sqrt[4]{7} , \zeta) = K$ where $\displaystyle \zeta = e^{\frac{2i \pi}{4}}=i$ . Now this extension has degree $\displaystyle 8$ since $\displaystyle 2, 4\vert [K:F]$ and $\displaystyle [K:f] \leq 8$. By the isomorphis extension theorem there are $\displaystyle \sigma , \tau: K\longrightarrow K$ automorphisms such that $\displaystyle \sigma (\sqrt[4]{7}) = i\sqrt[4]{7}$ and $\displaystyle \sigma (i) =i$ and $\displaystyle \tau (\sqrt[4]{7}) = \sqrt[4]{7}$ $\displaystyle \tau (i)=-i $

from this we know that $\displaystyle G=Gal(\frac{K}{F}) \cong D_8$ . Now the fixed field of $\displaystyle < \sigma>$ is $\displaystyle F(i)$ and the fixed field of $\displaystyle < \tau>= F(\sqrt[4]{7})$, but I'm having trouble with the other subgroups $\displaystyle <\sigma ^2>$ , $\displaystyle <\sigma \tau>$, $\displaystyle <\sigma ^2 \tau>$, $\displaystyle < \sigma ^3 \tau>$. I know they must be degree $\displaystyle 4$ extensions of $\displaystyle F$, but using the trace I get either trivial fixed elements or ones that I don't know how to check their degree over $\displaystyle F$.

Tnaks in advance