Class number of cyclotomic extension

Question

Show that the class number of Q$(\zeta_m)$ divides the class number of Q$(\zeta_n)$ if $m$ divides $n$.

I've found this statement in some notes on class field theory, but I have no idea how to approach it.

Answer 1

You first prove that the maximal abelian subextension of the Hilbert class field of a full cyclotomic field is the cyclotomic field itself. This then implies the existence of an unramified abelian extension of ${\mathbb Q}(\zeta_n)$ with degree $h({\mathbb Q}(\zeta_m))$.