Suppose I start with a number field $F$ and a character $$ \chi :G_F =\operatorname{Gal}(\overline F/F)\rightarrow {\overline {\mathbf Q}}^\times. $$
How does one get a character on ideals of $F$? What does conductor being $\mathfrak n$ mean?
I know that $\chi $ factors through some finite Galois extension $H$.
A guess of mine is that define $\chi (\mathfrak p)= \chi (\operatorname{Frob}_\mathfrak p )$ and conductor is product of all the primes that ramify in $H$ so $\chi $ of these primes is anyway $0$.
I was also given to understand that if $F$ is totally real then $H/F$ is cyclic. Why is this so?
Any help is appreciated, feel free to give any reference for this.
First of all, $\chi$ factors over through the abelianization $G_F^{ab}$, let the induced character on $G_F^{ab}$ also be denoted by $\chi$. We have by idelic global class field theory, a surjective continuous homomorphism from the idele class group $C_K \to G_F^{ab}$. (This is an inverse limit of Artin maps). Thus composing with this surjective homomorphism, we obtain a character of $C_K$, i.e. a Hecke character.