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I am trying to determine the number of subfields of $\mathbb Q (\zeta_{16})$ for which $[M :\mathbb Q]=2$ and their structure. (Where $M$ is a subfield and $\zeta_n$ is $n$th primitive root of unity )

It seems to be easier when $n$ is a prime but...

I believe the $Gal(\mathbb Q(\zeta_{16}):\mathbb Q)$ is has order 8 and the structure $(\mathbb Z/16\mathbb Z)^* ≅ C_2 \times C_4$, is this correct and could I use these facts to help me find the subfields? Would it have the structure of $C_2$ perhaps?

Would appreciate any help!

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    The quadratic subfields are $\mathbb{Q}(i),\mathbb{Q}(\cos(2\pi / 8))=\mathbb{Q}(\sqrt{2}), \mathbb{Q}(i\sin(2\pi / 8)) =\mathbb{Q}(i\cos(2\pi / 8)) = \mathbb{Q}(i \sqrt{2}) = \mathbb{Q}( \sqrt{-2})$ – reuns Nov 10 '17 at 13:07
  • By Galois theory, the subfields of $\mathbb Q(\zeta_n)$ are in bijection with the subgroups of the Galois group. – Arbutus Feb 05 '19 at 03:41

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