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

Let $K$ be a number field and $\mathcal{O}_1, \mathcal{O}_2$ two orders in $K$. If $\mathcal{O}_1 \subset \mathcal{O}_2$, then is it always true that there is a containment of ring class fields for $\mathcal{O}_1$ and $\mathcal{O}_2$ in the other direction?

I thought about this question in two ways, but neither quite gets through the hoop. First, there's a lemma that says that if there is a containment between kernels of the Artin maps of two Abelian extensions with respect to a fixed modulus, then there is a containment of fields. However, in this case the two maps will not have the same modulus.

I suppose that an affirmative answer to my question implies that if $\mathcal{O}_1 \subset \mathcal{O}_2$, then $C(\mathcal{O}_2)$ is a quotient of $C(\mathcal{O}_1)$. This isn't quite clear to me: you're modding out by more principal ideals, but aren't you starting out with a different group of fractional ideals?

share|cite|improve this question

Your Answer


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

Browse other questions tagged or ask your own question.