Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
What's the reason for the downvotes? Unless you make it clear, I can't improve my question. – 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: $N(Cl(K)) = Cl(k)$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.