Galois subfields and subgroups

Find the minimal Galois extension $L$ of $\mathbb{Q}$ containing $\mathbb{Q}(\sqrt[4]{5})$

$L=\mathbb{Q}(\sqrt[4]{5}, i)$ is a splitting field of $X^4-5$ over $\mathbb{Q}$

Describe the structure of $Gal(L/\mathbb{Q})$

$H=Gal(L/\mathbb{Q}(i))$ is a cyclic subgroup of order $4$. Consider automorphisms: $$\tau_1(\sqrt[4]{5})=i\sqrt[4]{5}$$ $$\tau_2(\sqrt[4]{5})=\sqrt[4]{5}$$

Now do we need to fins Galois extension $L^H$.

I am not sure how to proceed. Can you help?

Find all subfields of degree $2$ over $\mathbb{Q}$ in $L$

Once we know the automorphisms and Galois extensions, we can consider subgroups consisting of combinations of $e$, the idendity, $\tau_1$ and $\tau_2$

How do we then find the subfields? Thanks

• I believe you still need to find the Galois group over the ground field, which has degree 8. It's not too hard to see that this is $D_8$. From there you can just look at the subgroups of order 4, as these correspond to subfields of degree 2. – J.G May 30 '16 at 23:40

Hint: For the first question, to describe an element of $\operatorname{Gal}(L/{\bf Q})$, you only need to say where $\sqrt[4]{5}$ goes and where $i$ goes, and check that the choice leads to an automorphism. For the second question, just use the Galois correspondence.