# Additive structure of totally positive quadratic integers

## TL;DR

This paper characterizes the additive structure of totally positive integers in real quadratic fields, identifying generators, relations, and uniquely decomposable integers, and shows this structure uniquely determines the field.

## Contribution

It provides a complete description of the additive semigroup of totally positive integers in real quadratic fields using continued fractions, and proves this structure uniquely identifies the field.

## Key findings

- Generators and relations described via continued fractions
- Characterization of uniquely decomposable integers
- Semigroup determines the quadratic field

## Abstract

Let $K=\mathbb Q(\sqrt D)$ be a real quadratic field. We consider the additive semigroup $\mathcal O_K^+(+)$ of totally positive integers in $K$ and determine its generators (indecomposable integers) and relations; they can be nicely described in terms of the periodic continued fraction for $\sqrt D$. We also characterize all uniquely decomposable integers in $K$ and estimate their norms. Using these results, we prove that the semigroup $\mathcal O_K^+(+)$ completely determines the real quadratic field $K$.