Motivation: Let $\ell$ be an odd prime. There is a conductor-preserving correspondence between primitive Dirichlet characters of order $\ell$ and cyclic, degree $\ell$ number fields $K/\mathbb{Q}$.

The proof of this correspondence can be found in Chapter 3 of Washington's ``Introduction to Cyclotomic Fields".

Question: Let $k$ be a CM field. Is there a conductor-preserving correspondence between primitive Hecke characters of order $\ell$ and cyclic, degree $\ell$ extensions of $k$?

I have tried to construct a proof of this correspondence for Hecke characters using global class field theory and the theory of complex multiplication, however, I am not totally confident with my answer.

Any answers/references on the topic would be greatly appreciated.

This question is a duplicate from the one I posted on the Stackexchange, which can be found here.