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 $v\in S$.
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!
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.