Finite ramification support for radical extensions ($S=S_{\infty }\cup \{v:v(n)>0\}\cup \{v:v(a)\not\equiv 0(\mathrm{mod}n)\text{ for some }[a]\in \Delta \}$)

For a number field and finite subgroup $\Delta \subset K^{\ast }/K^{\ast n}$, adjoining all the indicated nth roots gives an extension unramified outside the displayed finite set. The valuations modulo $n$ are independent of representatives. At a finite place outside $S$, divide each radicand by a local nth power of a uniformizer to obtain a unit. Its polynomial has separable reduction because $n$ is invertible. A finite unramified extension containing all residue roots contains all lifted roots by Hensel's lemma. This proof does not require ${\mu }_n\subset K$.