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. 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. 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 π. 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. 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.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (8)

Related courses (1)

Related lectures (18)

Non-Archimedean ordered field

Real closed field

In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. A real closed field is a field F in which any of the following equivalent conditions is true: F is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in F if and only if it is true in the reals.

Surreal number

In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.

MATH-417: Number theory II.b - selected topics

This year's topic is "Adelic Number Theory" or how the language of adeles and ideles and harmonic analysis on the corresponding spaces can be used to revisit classical questions in algebraic number th

Properties of Real Numbers MATH-101(g): Analysis I

Explains the properties of subsets of real numbers, including Supremum, Infimum, intervals, open sets, and closed sets.

Complex Numbers and 2x2 Matrices MATH-101(de): Analysis I (German)

Covers complex numbers, quadratic equations, solutions in C, and 2x2 matrices.

Q is a Field MOOC: Analyse I

Covers the properties of rational numbers and introduces the concept of an ordered field in Q.