Example 1

There exists a largest abelian extension of exponent dividing $n$ of K, contained in $\Omega$ ; it is obtained by adjoining to K the n-th roots of all its elements ; its Galois group may be identified with $\operatorname{Hom}(K^*/K^{*n}, \mu_n(K))$, hence also with $\operatorname{Hom}(K^*, \mu_n(K))$.