On the class number of a cyclotomic number field of an odd prime order Is the following proposition true? If yes, how would you prove this?
Proposition
Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th  root of unity in $\mathbb{C}$.
Let $K = \mathbb{Q}(\zeta)$.
Let $k$ be a subfield of $K$.
Let $h_0$ be the class number of $k$.
Let $h$ be the class number of $K$.
Then $h$ is divisible by $h_0$.
Motivation
Let $k$ be the unique quadratic subfield of $K$.
The class number of $k$ can be relatively easily calculated if the discriminant of $k$ is small.
Hence, by the proposition, we can get useful information of the class number of $K$.
Effort
I considered the Hilbert class field $L/k$ and tried to use this.

Let $k^1$ denote the Hilbert class field of $k$. Since $K/k$ is completely ramified, $k^1/k$ is disjoint from $K/k$, so class field theory predicts that the norm is surjective on ideal classes: $N(Cl(K)) = Cl(k)$. 
