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If $L|Q$ is an abelian extension of Q, then use the theory of higher ramification groups to show that L lies in a succession of $L_n(\zeta_n)$ where $L_n$ is a subfield of $L$ and hence we can make the ramification of $L_n$ smaller in exchange for adjoining appropriate roots of unity, and in the end obtain an unramified extension of Q, i.e. Q itself and hence $Q(\zeta_n)$ contains $L$ for some natural integer n.