Question

The well-known Kronecker-Weber theorem.
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.
I must say that this is not my idea, instead, it is contained in here
which is by Keith Conrad.
And if @K Conrad is upset about what I do, then I will delete my post then and there.