Comparing ideal class numbers of different orders - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:22:44Z http://mathoverflow.net/feeds/question/112484 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112484/comparing-ideal-class-numbers-of-different-orders Comparing ideal class numbers of different orders Oblomov 2012-11-15T14:18:41Z 2012-11-16T08:55:28Z <p>Let $P$ be a monic irreducible integral polynomial. Let $K=\mathbf Q[X]/(P)$ be the associated number field, $\mathcal O$ be its ring of integers and $R$ be the order $\mathbf Z[X]/(P)$. (In general, $\mathcal O$ and $R$ do not coincide.) Both $R$ and $\mathcal O$ have finite number of ideal classes.</p> <p>My question is: how do these class numbers compare?</p> <p>It seems to me that $\cdot \otimes_{R}\mathcal O$ gives a natural map from ideal classes in $R$ to ideal classes in $\mathcal O$ which is surjective if I am not wrong. But, for example, in the case where the class number of $\mathcal O$ is 1, can we control the class number in $R$?</p> <p>Also: I know how to make pari/gp compute the class number in $\mathcal O$. Is there a way to have it compute the class number in $R$?</p>