Definition 15.108.1. We say that $A \to B$ or $A \subset B$ is an extension of discrete valuation rings if $A$ and $B$ are discrete valuation rings and $A \to B$ is injective and local. In particular, if $\pi _ A$ and $\pi _ B$ are uniformizers of $A$ and $B$, then $\pi _ A = u \pi _ B^ e$ for some $e \geq 1$ and unit $u$ of $B$. The integer $e$ does not depend on the choice of the uniformizers as it is also the unique integer $\geq 1$ such that

$\mathfrak m_ A B = \mathfrak m_ B^ e$

The integer $e$ is called the ramification index of $B$ over $A$. We say that $B$ is weakly unramified over $A$ if $e = 1$. If the extension of residue fields $\kappa _ A = A/\mathfrak m_ A \subset \kappa _ B = B/\mathfrak m_ B$ is finite, then we set $f = [\kappa _ B : \kappa _ A]$ and we call it the residual degree or residue degree of the extension $A \subset B$.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09E4. Beware of the difference between the letter 'O' and the digit '0'.