The lattice of primary ideals of orders in quadratic number fields

TL;DR

This paper characterizes the structure of primary ideals in orders of quadratic number fields where the conductor is a prime ideal, revealing a layered lattice structure that varies with the prime's splitting behavior.

Contribution

It provides a complete description of the lattice of primary ideals in quadratic orders with prime conductor, detailing the structure based on prime splitting.

Findings

Lattice of primary ideals has a layered structure.

Structure varies depending on whether the prime is split, inert, or ramified.

Identifies the core layer of the lattice as ideals not contained in the square of the conductor.

Abstract

Let $O$ be an order in a quadratic number field $K$ with ring of integers $D$, such that the conductor $\mathfrak F = f D$ is a prime ideal of $O$, where $f\in\mathbb Z$ is a prime. We give a complete description of the $\mathfrak F$-primary ideals of $O$. They form a lattice with a particular structure by layers; the first layer, which is the core of the lattice, consists of those $\mathfrak F$-primary ideals not contained in $\mathfrak F^2$. We get three different cases, according to whether the prime number $f$ is split, inert or ramified in $D$.