Class number divisibility under total ramification ($h_L\mid h_M$)

If $M/L$ is an extension of number fields, $L$ has no real places, and one finite prime of $L$ is totally ramified in $M$, then $h_L\mid h_M$. The Hilbert class field $H_L$ has trivial intersection with $M$: any common intermediate extension would be both unramified and totally ramified at that prime. Thus $M{H}_L/M$ is an unramified abelian extension of degree $h_L$, and is contained in $H_M$.