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In mathematics, a '''non-Archimedean ordered field''' is an [[ordered field]] that does not satisfy the [[Archimedean property]].  Examples are the [[Levi-Civita field]], the [[hyperreal number]]s, the [[surreal number]]s, the [[Dehn planes|Dehn field]], and the field of [[rational function]]s with real coefficients with a suitable order.

==Definition==
The [[Archimedean property]] is a property of certain ordered fields such as the [[rational number]]s or the [[real number]]s, stating that every two elements are within an integer multiple of each other. If a field contains two positive elements {{math|''x'' < ''y''}} for which this is not true, then {{math|''x''/''y''}} must be an infinitesimal, greater than zero but smaller than any integer [[unit fraction]]. Therefore, the negation of the Archimedean property is equivalent to the existence of infinitesimals.

==Applications==
[[Hyperreal number|Hyperreal field]]s, non-Archimedean ordered fields containing the real numbers as a subfield, may be used to provide a mathematical foundation for [[non-standard analysis]].

[[Max Dehn]] used the Dehn field, an example of a non-Archimedean ordered field, to construct [[non-Euclidean geometry|non-Euclidean geometries]] in which the [[parallel postulate]] fails to be true but nevertheless triangles have angles summing to {{math|π/2}}.<ref>{{Citation | last1=Dehn | first1=Max | author1-link=Max Dehn | title=Die Legendre'schen Sätze über die Winkelsumme im Dreieck | url=http://books.google.com/books?id=vEbWAAAAMAAJ&pg=PA404 | doi=10.1007/BF01448980 | jfm=31.0471.01 | year=1900 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=53 | issue=3 | pages=404–439}}.</ref>{{Dubious|Dehn's counterexample|date=February 2012}}

The field of rational functions over <math>\R</math> can be used to construct an ordered field which is [[Cauchy complete|complete]] (in the sense of convergence of Cauchy sequences) but is not the real numbers.<ref>''Counterexamples in Analysis'' by Bernard R. Gelbaum and John M. H. Olmsted, Chapter 1, Example 7, page 17.</ref>  This completion can be described as the field of [[Formal_power_series#Formal_Laurent_series|formal Laurent series]] over <math>\R</math>.  Sometimes the term complete is used to mean that the [[Least-upper-bound property|least upper bound property]] holds.  With this meaning of [[Dedekind-complete|complete]] there are no complete non-Archimedean ordered fields.  The subtle distinction between these two uses of the word complete is occasionally a source of confusion.

==References==
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[[Category:Ordered algebraic structures]]
[[Category:Real algebraic geometry]]
[[Category:Non-standard analysis]]