# A complete classification of well-rounded real quadratic ideal lattices

## TL;DR

This paper classifies all well-rounded ideal lattices from real quadratic fields, identifying specific divisors of the discriminant that produce such lattices, thus completing the understanding of their structure.

## Contribution

It provides a complete characterization of well-rounded ideal lattices in real quadratic fields based on divisors of the discriminant, a novel classification result.

## Key findings

- Identifies divisors of the discriminant corresponding to well-rounded lattices
- Establishes inequalities relating divisors to the discriminant for well-roundedness
- Completes the classification of such lattices in real quadratic fields

## Abstract

We provide a complete classification of well-rounded ideal lattices arising from real quadratic fields. We show that the ideals that give rise to such lattices are precisely the ones that correspond to divisors $a$ of the discriminant $d$ that satisfy $\sqrt{\frac{d}{3}}<a<\sqrt{3d}.$