nLab archimedean integral domain

Idea

An archimedean integral domain is an ordered integral domain that satisfies the archimedean property.

Properties

Every archimedean integral domain extension of the integers $\mathbb{Z}[x]$ is a dense linear order.

Since $\mathbb{Z}[x]$ is archimedean, it does not have either infinite or infinitesimal elements, which means there exists an integer $a$ and natural number $b$ such that $0 \lt x - a \lt 1$ and $1 \lt b(x - a)$. $0 \lt x - a \lt 1$ implies $0 \lt (d - c)(x - a) \lt d - c$ and $c \lt (d - c)(x - a) + c \lt d$ for all $c$ and $d$ in $\mathbb{Z}[x]$. Let $y = (d - c)(x - a) + c$. Since there exists an element $y$ such that $c \lt y \lt d$ for all $c$ and $d$, $\mathbb{Z}[x]$ is a dense linear order.

This means that the Dedekind completion of every archimedean integral domain extension of the integers is the integral domain of real numbers.

Examples

Archimedean integral domains include

• integers
• rational numbers
• decimal rational numbers
• dyadic rational numbers
• real numbers

Non-archimedean integral domains include

• p-adic integers.

Related concepts

• archimedean field

• archimedean group

• archimedean difference protoring

• enriched setoid