Ramification of ideals through ring/algebra homomorphisms is the dual incarnation of branch points of branched covering spaces.
Given a ring injection $i \colon B \to A$ (for instance the inclusion of $\mathcal{O}_K$ in $\mathcal{O}_L$, where $K \to L$ is a field extension and $\mathcal{O}_K$ (resp. $\mathcal{O}_L$) is the ring of integers of $K$ (resp. $L$)), then a prime ideal $J \subset B$ is said to be ramified in $A$ if $\iota(J) A\subset A$ is not a prime ideal anymore.
Here $\iota(J)A$ will be a product of powers of prime ideals of $A$, and the ramification index of $J$ at a prime ideal of $A$ is the power with which this appears.
Kummer's theorem?
Last revised on March 1, 2021 at 00:35:28. See the history of this page for a list of all contributions to it.