ring of integers

Last updated January 27, 2022

Operations

( norm) Suppose $I$ is an $\mathcal{O}K$- fractional ideal in $K$. Then its norm is $$N{K/k}(I)\coloneqq(\mathcal{O}K:I)\mathcal{O}$$

$N_{K/k}:\mathcal{F}(\mathcal{O}_K)\to\mathcal{F}(\mathcal{O})$ is a group homomorphism. (see dedekind domain]])
Suppose $P_x$ is Eisenstein wrt $p$. Then $p\nmid (\mathcal{O}_K:\mathbb{Z} )$.
Let the bar symbol denote elements of $\mathcal{O}_K$ modulo $\mathfrak{B}$.
(reduction of homomorphisms)
1. Suppose $K/k$ and $K'/k$ are finite separable extensions, $\sigma:K/k\to K'/k$ a homomorphism. Then the map $\sigma_\mathfrak{p}:\mathcal{O}_K/\mathfrak{B}\to\mathcal{O}_K'/\mathfrak{B}'$ defined as $\sigma_\mathfrak{p}(x\mod\mathfrak{B})\coloneqq\sigma(x)\mod\mathfrak{B}'$ is a correctly defined homomorphism of extensions $K_\mathfrak{B}/k_\mathfrak{p}\to K'_{\mathfrak{B}'}/k_\mathfrak{p}$.
2. Reduction commutes with composition of homomorphisms, so reduction of an isomorphism is an isomorphism.