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

In terms of a subfield of the reals

An Archimedean field is a field that is isomorphic to a subfield of the field of real numbers.

In terms of a total ordering

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

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

Relation with other properties

Weaker properties

• Formally real field: For proof of the implication, refer Archimedean implies formally real and for proof of its strictness (i.e. the reverse implication being false) refer Formally real not implies Archimedean.
• Field of characteristic zero

Incomparable properties

• Euclidean field
• Pythagorean field