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Encyclopedia > Ordered field

In mathematics, an ordered field is a field together with an ordering of its elements. This concept was introduced by Emil Artin in 1927. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ...

Contents

Definition

There are two equivalent definitions, depending on which properties one takes as the definition for an ordered field.


Def 1: A total order on F

A field (F,+,*) together with a total order ≤ on F is an ordered field if the order satisfies the following properties: In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...

  • if ab then a + cb + c
  • if 0 ≤ a and 0 ≤ b then 0 ≤ a b

It follows from these axioms that for every a, b, c, d in F:

  • Either −a ≤ 0 ≤ a or a ≤ 0 ≤ −a.
  • We are allowed to "add inequalities": If ab and cd, then a + cb + d
  • We are allowed to "multiply inequalities with positive elements": If ab and 0 ≤ c, then acbc.

Def 2: An ordering on F

An ordering of a field F is a subset PF that has the following properties:

  • F is the disjoint union of P, −P, and the element 0. That is, for each xF, then exactly one of the following conditions is true: x = 0, xP or −xP.
  • For x and y in P, both x+y and xy are in P.

The subset P are called the positive elements of F.


We next define x < y to mean that y − xP (so that y − x > 0 in a sense). This relation satisfies the expected properties:

  • If x < y and y < z, then x < z. (transitivity)
  • If x < y and z > 0, then xz < yz.
  • If x < y and x,y > 0, then 1/y < 1/x

The statement xy will mean that either x < y or x = y. In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ...


Properties of ordered fields

  • 1 is positive. (Justification: either 1 is positive or −1 is positive. If −1 is positive, then (−1)(−1) is positive, which is a contradiction)
  • An ordered field has characteristic 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc. If the field had characteristic p > 0, then −1 would be the sum of p − 1 ones, but −1 is not positive). In particular, finite fields cannot be ordered.
  • Squares are non-negative. 0 ≤ a2 for all a in F. (Follows by a similar argument to 1 > 0)

Every subfield of an ordered field is also an ordered field (inheriting the induced ordering). The smallest subfield is isomorphic to the rationals (as for any other field of characteristic 0), and the order on this rational subfield is the same as the order of the rationals themselves. If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be Archimedean. For example, the real numbers form an Archimedean field, but every hyperreal field is non-Archimedean. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ...


The following conditions are equivalent when R is an ordered field.

  1. R is a real closed field.
  2. R is a maximal ordered field.
  3. R is a maximal real field.

In mathematics, a real closed field is an ordered field F in which any of the following equivalent conditions are true: Every non-negative element of F has a square root in F, and any polynomial of odd degree with coefficients in F has at least one root in F...

Topology induced by the order

If F is equipped with the order topology arising from the total order ≤, then the axioms guarantee that the operations + and * are continuous, so that F is a topological field. In mathematics, the order topology is a topology that can be defined on any totally ordered set. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R &#8594; R, where R × R carries the product topology. ...


Examples of ordered fields

Examples of ordered fields are:

The surreal numbers form a proper class rather than a set, but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal numbers. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form a0xn + a1xn&#8722;1 + ··· + an &#8722;1x + an = 0 where n is a positive integer called... In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a rational function is a ratio of polynomials. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In mathematics, a real closed field is an ordered field F in which any of the following equivalent conditions are true: Every non-negative element of F has a square root in F, and any polynomial of odd degree with coefficients in F has at least one root in F... The superreal numbers compose a more inclusive category than hyperreal number. ... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similiar to superreal numbers and hyperreal numbers. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...


Which fields can be ordered?

Every ordered field is a formally real field, i.e., 0 cannot be written as a sum of nonzero squares. In mathematics, a formally real field in field theory is a field that shares certain algebraic properties with the real number field. ...


Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order is often not uniquely determined.)


Finite fields cannot be turned into ordered fields, because they do not have characteristic 0. The complex numbers also cannot be turned into an ordered field, as −1 is a square (of the imaginary number i) and would thus be positive. Also, the p-adic numbers cannot be ordered, since Q2 contains a square root of −7 and Qp (p > 2) contains a square root of 1 − p. In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... The title given to this article is incorrect due to technical limitations. ...


References

  • Lang, Serge (1997). Algebra, 3rd ed., reprint w/ corr., Addison-Wesley. ISBN 978-0-201-55540-0. 

  Results from FactBites:
 
Field (mathematics) - Wikipedia, the free encyclopedia (1581 words)
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers.
For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F.
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