Galois extension, lattice of subfields

1. Dec 10, 2008

mathsss2

$$K=\mathbb{Q}(\sqrt{2+\sqrt{2}})$$ is a Galois extension of $$\mathbb{Q}$$ [I showed this]. Determine $$\text{Gal}(K/\mathbb{Q})$$ and describe the lattice of subfields $$\mathbb{Q} \subset F \subset K$$.

I found that $$\text{Gal}(K/\mathbb{Q})=\mathbb{Z}_4$$. I do not know how to draw the lattice of subfields, so far I have this: [see attachment], but I do not think it is right. I need help with the lattice of subfields.

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2. Dec 10, 2008

Office_Shredder

Staff Emeritus
Recall that the lattice of subfields is directly correlated to the lattice of subgroups of the galois group. This should make it easier