This question is a sort of follow up to this question, where I introduced context.
Neukirch mysterious homomorphism in Abstract Kummer Theory (in his book ANT)
The thing I don't understand know is how an element $\chi$ in $\operatorname{Hom}(G(L/K,\mu_{\wp})$ "defines" a cyclic extension. What does he mean by that?
In general, by the Galois correspondance, a surjective group morphism $f: Gal(L/K)\to G$ defines a Galois subextension of $L/K$ with Galois group $G$, by $L^{\ker(f)}$.
Since $\mu_\wp$ is a finite subgroup of $K^*$, it is cyclic. So for any $\chi: G(L/K)\to \mu_\wp$, the image of $\chi$ is a cyclic group, so $\chi$ defines a cyclic subextension of $L/K$.