# 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 non-standard 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 π/2.[1][dubious ]

The field of rational functions over $\R$ 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 $\R$. 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. ^ .
2. ^ Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted, Chapter 1, Example 7, page 17.