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I am reading J. S. Milne's Class Field Theory and have a question about his proof for Lemma 7.3, Chapter VII https://www.jmilne.org/math/CourseNotes/CFT.pdf:

Lemma 7.3: Given a number field $K$, a finite set $S$ of finite primes of $K$. and an integer $m>0$, there exists a totally complex cyclic cyclotomic extension $L$ of $K$ such that $m| [L^v:K_v]$ for all $v\in S$.

Proof of Lemma 7.3

I did not see how to construct the totally complex extension, and actually the extension constructed in the proof above $\mathbb{Q}$ is totally real. But I do need the extension to be totally complex in order to finish the proof of Proposition 7.2.

Any help will be appreciated!

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    $\begingroup$ "actually the extension constructed in the proof above $\mathbb Q$ is totally real" What makes you think it is totally real? Are you suggesting that Milne's proof of lemma 7.3 is wrong? $\endgroup$
    – Suzet
    Commented Jan 5, 2023 at 23:30
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    $\begingroup$ For example, let $L=L(l^r)$ over $\mathbb{Q}$, where $l$ is an odd prime. Then $L$ is a cyclic extension of prime order over $\mathbb{Q}$. And any extension of prime degree over $\mathbb{Q}$ is totally real. $\endgroup$ Commented Jan 5, 2023 at 23:38
  • $\begingroup$ @Kris Could you please explain why does the extension need to be totally complex to finish the proof of Proposition 7.2? $\endgroup$
    – debanjana
    Commented Jun 9, 2023 at 15:32

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Here is my solution to fix that proof:

We have got a cyclic extension $L/\mathbb{Q}$ satisfying the conditions on $m$. Then we can embedd $L$ into a larger totally complex cyclic extension, which still satisfies our conditions on $m$. For example, we can add a root of unity $\xi_n$ into $L$, where $n$ is large enough and such that the extension $L(\xi_n)/\mathbb{Q}$ is still cyclic (coprimes degrees). And $L(\xi_n)$ is totally complex clearly.

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