nLab
archimedean integral domain

Contents

Contents

Idea

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

Properties

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

Since [x]\mathbb{Z}[x] is archimedean, it does not have either infinite or infinitesimal elements, which means there exists an integer aa and natural number bb such that 0<xa<10 \lt x - a \lt 1 and 1<b(xa)1 \lt b(x - a). 0<xa<10 \lt x - a \lt 1 implies 0<(dc)(xa)<dc0 \lt (d - c)(x - a) \lt d - c and c<(dc)(xa)+c<dc \lt (d - c)(x - a) + c \lt d for all cc and dd in [x]\mathbb{Z}[x]. Let y=(dc)(xa)+cy = (d - c)(x - a) + c. Since there exists an element yy such that c<y<dc \lt y \lt d for all cc and dd, [x]\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

Non-archimedean integral domains include

Created on May 27, 2021 at 05:57:34. See the history of this page for a list of all contributions to it.