Archimedean field

This article defines a field property: a property that can be evaluated to true/false for any field.
View a complete list of field properties|View a complete list of field extension properties

Definition

An Archimedean field is a field with a total ordering $\le$ on its elements satisfying the following:

• $0\le 1$.
• $a\le b⟹-b\le -a$.
• $a\le b$ and $c\le d$ implies that $a+c\le b+d$.
• $0\le a$ and $b\le c$ implies that $ab\le ac$.
• The Archimedean property: For any $x\in K$, there exists a natural number $n$ such that $x\le n$, where the natural number $n$ is viewed as the element of $K$ obtained by adding $1$ to itself $n$ times.

Relation with other properties

Stronger properties

• Totally real field

Weaker properties

• Formally real field
• Field of characteristic zero