Non-Archimedean Ordered Field
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Non-Archimedean Ordered Field

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 numbers, the surreal numbers, the Dehn field, and the field of rational functions with real coefficients with a suitable order.

## Definition

The Archimedean property is a property of certain ordered fields such as the rational numbers or the real numbers, stating that every two elements are within an integer multiple of each other. If a field contains two positive elements x < y for which this is not true, then 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 fields, non-Archimedean ordered fields containing the real numbers as a subfield, may be used to provide a mathematical foundation for nonstandard analysis.

Max Dehn used the Dehn field, an example of a non-Archimedean ordered field, to construct non-Euclidean geometries in which the parallel postulate fails to be true but nevertheless triangles have angles summing to ?.[1][dubious ]

The field of rational functions over ${\displaystyle \mathbb {R} }$ can be used to construct an ordered field which is complete (in the sense of convergence of Cauchy sequences) but is not the real numbers.[2] This completion can be described as the field of formal Laurent series over ${\displaystyle \mathbb {R} }$. Sometimes the term complete is used to mean that the least upper bound property holds. With this meaning of 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

1. ^ Dehn, Max (1900), "Die Legendre'schen Sätze über die Winkelsumme im Dreieck", Mathematische Annalen, 53 (3): 404-439, doi:10.1007/BF01448980, ISSN 0025-5831, JFM 31.0471.01.
2. ^ Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted, Chapter 1, Example 7, page 17.

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