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.