Ordered field
Encyclopedia
In mathematics
, an ordered field is a field
together with a total order
ing of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real number
s, by mathematicians including David Hilbert
, Otto Hölder
and Hans Hahn
. In 1926, this grew eventually into the Artin–Schreier theory of ordered fields and formally real field
s.
An ordered field necessarily has characteristic
0, i.e., the elements 0, 1, , , … are all different. This implies that an ordered field necessarily contains an infinite number of elements. Finite field
s cannot be ordered.
Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic
to the rational number
s. Any Dedekind-complete ordered field is isomorphic to the real numbers. Squares are necessarily non-negative in an ordered field. This implies that the complex number
s cannot be ordered since the square of the imaginary unit
i is -1. Every ordered field is a formally real field.
(F,+,*) together with a total order
≤ on F is an ordered field if the order satisfies the following properties:
P ⊂ F that has the following properties:
If in addition, the subset F is the union of P and −P, we call P a positive cone of F.
The nonzero elements of P are called the positive elements of F.
An ordered field is a field F together with a positive cone P.
Given a field ordering ≤ as in Def 1, the elements such that x≥0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤P by setting x≤y to mean y − x ∈ P. This total ordering ≤P satisfies the properties of Def 1.
For every a, b, c, d in F:
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
. Otherwise, such field is a non-Archimedean ordered field
and contains infinitesimal
s. For example, the real number
s form an Archimedean field, but every hyperreal field is non-Archimedean.
An ordered field K is the real number field if it satisfies the axiom of Archimedes and every Cauchy sequence
of K converges within K.
arising from the total order ≤, then the axioms guarantee that the operations + and * are continuous, so that F is a topological field.
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.
, i.e., 0 cannot be written as a sum of nonzero squares.
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 field
s cannot be turned into ordered fields, because they do not have characteristic 0. The complex number
s 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an ordered field is a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
together with a total order
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
ing of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, by mathematicians including David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, Otto Hölder
Otto Hölder
Otto Ludwig Hölder was a German mathematician born in Stuttgart.Hölder first studied at the Polytechnikum and then in 1877 went to Berlin where he was a student of Leopold Kronecker, Karl Weierstraß, and Ernst Kummer.He is famous for many things including: Hölder's inequality, the Jordan–Hölder...
and Hans Hahn
Hans Hahn
Hans Hahn was an Austrian mathematician who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.-Biography:...
. In 1926, this grew eventually into the Artin–Schreier theory of ordered fields and formally real field
Formally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that admits an ordering which makes it an ordered field.-Alternative Definitions:...
s.
An ordered field necessarily has characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
0, i.e., the elements 0, 1, , , … are all different. This implies that an ordered field necessarily contains an infinite number of elements. Finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s cannot be ordered.
Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s. Any Dedekind-complete ordered field is isomorphic to the real numbers. Squares are necessarily non-negative in an ordered field. This implies that the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s cannot be ordered since the square of the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
i is -1. Every ordered field is a formally real field.
Definition
There are two equivalent definitions of an ordered field. Def 1 appeared first historically and is a first-order axiomatization of the ordering ≤ as a binary predicate. Artin and Schreier gave Def 2 in 1926, which axiomatizes the subcollection of nonnegative elements. It subcollection is termed a positive cones (Def 2 below) in 1926. Although Def 2 is higher-order, viewing positive cones as maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.Def 1: A total order on F
A fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
(F,+,*) together with a total order
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
≤ on F is an ordered field if the order satisfies the following properties:
- if a ≤ b then a + c ≤ b + c
- if 0 ≤ a and 0 ≤ b then 0 ≤ a b
Def 2: A positive cone of F
A prepositive cone of a field F is a subsetSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
P ⊂ F that has the following properties:
- For x and y in P, both x+y and xy are in P.
- If x is in F, then x2 is in P.
- The element −1 is not in P.
If in addition, the subset F is the union of P and −P, we call P a positive cone of F.
The nonzero elements of P are called the positive elements of F.
An ordered field is a field F together with a positive cone P.
Equivalence of the two definitions
Let F be a field. There is a bijection between the field orderings of F and the positive cones of F.Given a field ordering ≤ as in Def 1, the elements such that x≥0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤P by setting x≤y to mean y − x ∈ P. This total ordering ≤P satisfies the properties of Def 1.
Properties of ordered fields
- 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
For every a, b, c, d in F:
- Either −a ≤ 0 ≤ a or a ≤ 0 ≤ −a.
- We are allowed to "add inequalities": If a ≤ b and c ≤ d, then a + c ≤ b + d
- We are allowed to "multiply inequalities with positive elements": If a ≤ b and 0 ≤ c, then ac ≤ bc.
- 1 is positive. (Proof: either 1 is positive or −1 is positive. If −1 is positive, then (−1)(−1) = 1 is positive, which is a contradiction)
- An ordered field has characteristicCharacteristic (algebra)In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
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
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to the rationals
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
(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
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 ordered or normed groups, fields, and other algebraic structures. Roughly speaking, it is the property of having no infinitely large or...
. Otherwise, such field is a non-Archimedean ordered field
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...
and contains infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s. For example, the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s form an Archimedean field, but every hyperreal field is non-Archimedean.
An ordered field K is the real number field if it satisfies the axiom of Archimedes and every Cauchy sequence
Cauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
of K converges within K.
Topology induced by the order
If F is equipped with the order topologyOrder topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...
arising from the total order ≤, then the axioms guarantee that the operations + and * are continuous, so that F is a topological field.
Examples of ordered fields
Examples of ordered fields are:- the rational numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s - the real algebraic numbers
- the computable numberComputable numberIn mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm...
s - the real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s - the field of real rational functions
, where p(x) and q(x), 
are polynomialPolynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s with real coefficients, can be made into an ordered field where the polynomial p(x) = x is greater than any constant polynomial, by defining that
whenever 
, for 
. This ordered field is not Archimedean. - The field of formal Laurent seriesFormal power seriesIn mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
with real coefficients
, where x is taken to be infinitesimal and positive - real closed fieldReal closed fieldIn 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.-Definitions:...
s - superreal numberSuperreal numberIn abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theory, and the study of Banach algebras...
s - hyperreal numberHyperreal numberThe system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
s
The surreal numbers form a proper class
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
rather than a set, but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal numbers.
Which fields can be ordered?
Every ordered field is a formally real fieldFormally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that admits an ordering which makes it an ordered field.-Alternative Definitions:...
, i.e., 0 cannot be written as a sum of nonzero squares.
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 field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s cannot be turned into ordered fields, because they do not have characteristic 0. The complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s 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.