In mathematics, a non-Archimedean ordered field is an
ordered field that does not satisfy the
Archimedean property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
The property, typi ...
. 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
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
The property, typi ...
is a property of certain ordered fields such as the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s or the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, stating that every two elements are within an integer multiple of each other. If a field contains two positive elements for which this is not true, then 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 .
The field of rational functions over
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.
[''Counterexamples in Analysis'' by Bernard R. Gelbaum and John M. H. Olmsted, Chapter 1, Example 7, page 17.] This completion can be described as the field of
formal Laurent series over
. 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
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Ordered algebraic structures
Real algebraic geometry
Nonstandard analysis