Proposition 11.3. Assuming: - $K$ a number field - ${\mu }_n\subset K$ - $S$ a finite set of promes of $K$ Then there are only finitely many extensions $L∕K$ such that

• (i) $L∕K$ is a finite abelian extension of exponent dividing $n$
• (ii) $L∕K$ is unramified at all $𝔭\notin S$