Existence of a totally complex cyclic cyclotomic extension

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!

• "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? Commented Jan 5, 2023 at 23:30
• 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. Commented Jan 5, 2023 at 23:38
• @Kris Could you please explain why does the extension need to be totally complex to finish the proof of Proposition 7.2? Commented Jun 9, 2023 at 15:32

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.