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.

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    $\begingroup$ What's the reason for the downvotes? Unless you make it clear, I can't improve my question. $\endgroup$ – Makoto Kato Jul 27 '12 at 21:05

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: $N(Cl(K)) = Cl(k)$.

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