Abelian extensions and characters I do not understand the following argument by Marcus:
Let $K|\mathbb Q$ an abelian extension where $K$ is a number field. We identify $\mathbb Z_m^*$ with the Galois group of $\mathbb Q(\omega)$ over $\mathbb Q$.[This is fine]
Then $G$ is a homomorphic image of $\mathbb Z_m^*$.[What does it mean? It means maybe that $G$ is isomorphic to a quotient of $\mathbb Z_m^*$? Because this would be fine for me.]
Hence characters of $G$ can then be regarded as characters $mod$ $m$.[Why? What does it mean?]
Thus we consider the group of characters of $G$ as a subgroup of the group of characters of $\mathbb Z_m^*$. [why? Because if $G$ is isomorphic to subgroup of $\mathbb Z_m^*$ this is fine]
 A: I guess that it is assumed that $K$ is a subfield of $\mathbb{Q}[\zeta_m]$. Since the Galois group of $\mathbb{Q}[\zeta_m]/\mathbb{Q}$ is $\mathbb{Z}_m^*$, the Galois group of the subextension $K/\mathbb{Q}$ is a homomorphic image of it, namely there is a surjection $Gal(\mathbb{Q}[\zeta_m]/\mathbb{Q})\to Gal(K/\mathbb{Q})$, which takes the Galois action on $\mathbb{Q}[\zeta_m]$ and restricts it to $K$. A character on $Gal(K/\mathbb{Q})$ can then be lifted to a character on $Gal(\mathbb{Q}[\zeta_m]/\mathbb{Q})$.
A: Let $S$ be a subgroup of a finite abelian group $T$.  We can identify $X(T/S)$ as a subgroup of $X(T)$: namely, $X(T/S)$ consists of those characters $\chi \in X(T)$ for which $\chi(s) = 1$ for all $s \in S$.  This follows from the fact that every character of $S$ extends to a character of $T$.
Now let $T = \textrm{Gal}(\mathbb{Q}(\omega)/\mathbb{Q})$, where $\omega$ is a primitive $m$th root of unity, and let $S = \textrm{Gal}(\mathbb{Q}(\omega)/K)$.  Then restriction $\sigma \mapsto \sigma|K$ induces an isomorphism of $T/S$ with the Galois group $\textrm{Gal}(K/\mathbb{Q})$.  Now we can identify the characters of $\textrm{Gal}(K/\mathbb{Q})$ with those characters of $T$ which are trivial on $S$.
