= Finite ramification support for radical extensions
{title2=$S=S_\infty\cup\{v:v(n)>0\}\cup\{v:v(a)\not\equiv0\pmod n\text{ for some }[a]\in\Delta\}$}
For a <number field> and finite subgroup $\Delta\subset K^*/K^{*n}$, 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 $\mu_n\subset K$.
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