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

I found that [tex]\text{Gal}(K/\mathbb{Q})=\mathbb{Z}_4[/tex]. 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.

I found that [tex]\text{Gal}(K/\mathbb{Q})=\mathbb{Z}_4[/tex]. 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.