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I have the 16-th primitive root of unity $\zeta$ and the extension $\mathbb Q(\zeta):\mathbb Q$, I want to find the $L$ subfields of this extension for which $[L:\mathbb Q]=2$.

Could anyone tell me what these subfields are and what their structure is? I've looked for examples online that manipulate the elements but I found them confusing, could I use the fact that I know $\zeta^2$ is a primitive 8-th root of unity and $\zeta^4$ is a primitive 4-th root of unity?

Help appreciated!

user1836257
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    See this question; or this one. – Dietrich Burde Nov 22 '17 at 21:24
  • @DietrichBurde I understand the second one a little more, I know the galois group is isomorphic to $\mathbb Z/16 \mathbb Z)^*$ and is of order 8 but I am struggling to apply their method to my case. Are you able to show me where to go from there? I would be so grateful – user1836257 Nov 22 '17 at 21:38
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    You should describe the group $(\Bbb Z/(16))^\times$ explicitly, as product of a cyclic group of order four and a group of order two. Then describe the subgroups of order four, and their quotients. Alternatively, you might try to understand $\zeta_{16}$ in depth: what is $\zeta^4$, and what is $\zeta^8$? – Lubin Nov 23 '17 at 02:09
  • I believe that the subgroups of order 4 are $C_2 \times C_2$ ${ 1} \times C_4 $ and $C_2 \times C_4 $, what are the subfields I can deduce from this? – user1836257 Nov 23 '17 at 12:46
  • Would $\mathbb Q(\zeta_8)$ be one of them? – user1836257 Nov 23 '17 at 12:53
  • I guess what Lubin was suggesting is that you should write down the elements of those subgroups in terms of where they send $\zeta$! Only then you have a chance to figure out the corresponding fixed fields. Note, there are isomorphic but non-identical subgroups, and they correspond to different intermediate fields. – Jyrki Lahtonen Nov 23 '17 at 16:28

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