push-down theorem on class numbers

As in the parent (http://planetmath.org/ClassNumberDivisibilityInExtensions) entry, given a number field $K$, the class number of $K$ is denoted by $h_K$.

Theorem (Pushing-Down Theorem).

Let $E\mathrm{/}F$ be a $p$-extension of number fields and suppose that only one prime ideal of $F$ is ramified in $E$ and that this prime is totally ramified. Then $p\mathrm{|}{h}_E$ implies $p\mathrm{|}{h}_F$.