residue class degree, ramification index

Last updated January 27, 2022

Definition

Suppose $\mathcal{O}_1\subset\mathcal{O}_2$ are dedekind domains, $\mathfrak{p}_1\subset\mathcal{O}_1$ and $\mathfrak{p}_2\subset\mathcal{O}_2$ are nonzero prime ideal such that $\mathfrak{p}_2$ lies over $\mathfrak{p}_1$ (i.e. $\mathfrak{p}_1=\mathfrak{p}_2\cap\mathcal{O}_1$).

• The residue class degree, or inertia index, is $$f(\mathfrak{p}_2/\mathfrak{p}1)\coloneqq [k{\mathfrak{p}2}:k{\mathfrak{p}_1}]$$ (notice nonzero prime ideals are maximal in Dedekind domains, so the above is the degree of the Field extension). We allow infinite values.
• The ramification index is the (necessarily finite) number $$e(\mathfrak{p}_2/\mathfrak{p}1)\coloneqq v{\mathfrak{p}_2}(\mathfrak{p}_1\mathcal{O}_2)$$

Properties

1. Suppose $\mathcal{O}_1\subset\mathcal{O}_2\subset\mathcal{O}_3$ and $\mathfrak{p}_3\mid \mathfrak{p}_2\mid\mathfrak{p}_1$. Then
   1. $f(\mathfrak{p}_3/\mathfrak{p}_1)=f(\mathfrak{p}_3/\mathfrak{p}_2)f(\mathfrak{p}_2/\mathfrak{p}_1)$ so long as two out of the three are finite
   2. $e(\mathfrak{p}_3/\mathfrak{p}_1)=e(\mathfrak{p}_3/\mathfrak{p}_2)e(\mathfrak{p}_2/\mathfrak{p}_1)$ so long as two out of the three are finite
   3. If $\mathcal{O}1=\mathcal{O}\mathfrak{p}$ is a discrete valuation ring, $\mathcal{O}2=\widehat{\mathcal{O}}\mathfrak{p}$ its completion. Then $$f(\mathfrak{p}\widehat{\mathcal{O}}\mathfrak{p}/\mathfrak{p})=e(\mathfrak{p}\widehat{\mathcal{O}}\mathfrak{p}/\mathfrak{p})=1.$$ (see associated p-adic algebra)