= Solution
For $K=\mathbb Q(\zeta)$, every <Galois automorphism> sends the primitive root $\zeta$ to another primitive $m$th <root of unity>, uniquely $\zeta^a$ for $a\in(\mathbb Z/m\mathbb Z)^\times$. The <Cyclotomic character of a finite cyclotomic extension> is
$$
\boxed{\chi_m:\operatorname{Gal}(K/\mathbb Q)\longrightarrow(\mathbb Z/m\mathbb Z)^\times,\qquad\sigma(\zeta)=\zeta^{\chi_m(\sigma)}}.
$$
If $\chi_m(\sigma)=a$ and $\chi_m(\tau)=b$, then $(\sigma\tau)(\zeta)=\sigma(\zeta^b)=\zeta^{ab}$, proving the <group homomorphism> property. Its <kernel> is trivial: an automorphism fixing $\zeta$ fixes the field it generates. Thus the character is <injective>. Here its domain is the finite cyclotomic <Galois group>; the corresponding map from an <absolute Galois group> need not be injective.
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