In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Such fields will contain infinitesimal and infinitely large elements, suitably defined.
Suppose is an ordered field. We say that satisfies the Archimedean property if, for every two positive elements and of, there exists a natural number such that . Here, denotes the field element resulting from forming the sum of copies of the field element, so that is the sum of copies of .
An ordered field that does not satisfy the Archimedean property is a non-Archimedean ordered field.
The fields of rational numbers and real numbers, with their usual orderings, satisfy the Archimedean property.
Examples of non-Archimedean ordered fields are the Levi-Civita field, the hyperreal numbers, the surreal numbers, the Dehn field, and the field of rational functions with real coefficients (where we define to mean that for large enough t).
In a non-Archimedean ordered field, we can find two positive elements and such that, for every natural number, . This means that the positive element is greater than every natural number (so it is an "infinite element"), and the positive element is smaller than for every natural number (so it is an "infinitesimal element").
Conversely, if an ordered field contains an infinite or an infinitesimal element in this sense, then it is a non-Archimedean ordered field.
Hyperreal fields, non-Archimedean ordered fields containing the real numbers as a subfield, are 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]
The field of rational functions over
\R
\R