Hecke characters and Conductors

Question

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.

Answer 1

Let $k$ be a number field. By class field theory, any Hecke character $\chi$ of $k$ of order $\ell$ determines a cyclic extension $k'/k$ of degree $\ell$. Moreover, the set of Hecke characters determining this cyclic extension $k'/k$ equals $\{\chi,\chi^2,\dots,\chi^{\ell-1}\}$. These $\ell-1$ Hecke characters have the same conductor $\mathfrak{q}$, and the discriminant of $k'$ over $k$ equals their product $\mathfrak{q}^{\ell-1}$. In fact the product of the corresponding $L$-functions $L(s,\chi),\dots,L(s,\chi^{\ell-1})$ equals $\zeta_{k'}(s)/\zeta_k(s)$.