In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an ordered field is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
together with a
total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
ing of its elements that is compatible with the field operations. The basic example of an ordered field is the field of
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, and every
Dedekind-complete
In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if eve ...
ordered field is isomorphic to the reals.
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
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to 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 ration ...
s.
Squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
are necessarily non-negative in an ordered field. This implies that the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s cannot be ordered since the square of the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
''i'' is .
Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s cannot be ordered.
Historically, the
axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
,
Otto Hölder
Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.
Early life and education
Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Christ ...
and
Hans Hahn. This grew eventually into the
Artin–Schreier theory
In mathematics, Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic ''p''. introduced Artin–Schreier theory for ex ...
of ordered fields and
formally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field.
Alternative definitions
The definition given above is ...
s.
Definitions
There are two equivalent common definitions of an ordered field. The definition of total order appeared first historically and is a first-order axiomatization of the ordering
as a
binary predicate. Artin and Schreier gave the definition in terms of positive cone in 1926, which axiomatizes the subcollection of nonnegative elements. Although the latter is higher-order, viewing positive cones as prepositive cones provides a larger context in which field orderings are partial orderings.
Total order
A
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
together with a
(strict) total order on
is an if the order satisfies the following properties for all
* if
then
and
* if
and
then
Positive cone
A or preordering of a field
is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
that has the following properties:
[Lam (2005) p. 289]
* For
and
in
both
and
are in
* If
then
In particular,
* The element
is not in
A is a field equipped with a preordering
Its non-zero elements
form a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of the multiplicative group of
If in addition, the set
is the union of
and
we call
a positive cone of
The non-zero elements of
are called the positive elements of
An ordered field is a field
together with a positive cone
The preorderings on
are precisely the intersections of families of positive cones on
The positive cones are the maximal preorderings.
[
]
Equivalence of the two definitions
Let be a field. There is a bijection between the field orderings of and the positive cones of
Given a field ordering ≤ as in the first definition, the set of elements such that forms a positive cone of Conversely, given a positive cone of as in the second definition, one can associate a total ordering on by setting to mean This total ordering satisfies the properties of the first definition.
Examples of ordered fields
Examples of ordered fields are:
* 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 ration ...
s
* the real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s
* any subfield of an ordered field, such as the real algebraic numbers
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
or computable number
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive ...
s
* the field of real rational functions
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
, where and are polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s with real coefficients, , can be made into an ordered field where the polynomial is greater than any constant polynomial, by defining that whenever , for and . This ordered field is not Archimedean.
* The field of formal Laurent series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
with real coefficients, where ''x'' is taken to be infinitesimal and positive
* the transseries In mathematics, the field \mathbb^ of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric functions to a much broader class of ob ...
* 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.
D ...
s
* the superreal number
In 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 theo ...
s
* the hyperreal number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
s
The surreal numbers
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surrea ...
form a proper class
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
rather than a set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal numbers.
Properties of ordered fields
For every ''a'', ''b'', ''c'', ''d'' in ''F'':
* Either −''a'' ≤ 0 ≤ ''a'' or ''a'' ≤ 0 ≤ −''a''.
* One can "add inequalities": if ''a'' ≤ ''b'' and ''c'' ≤ ''d'', then ''a'' + ''c'' ≤ ''b'' + ''d''.
* One can "multiply inequalities with positive elements": if ''a'' ≤ ''b'' and 0 ≤ ''c'', then ''ac'' ≤ ''bc''.
* Transitivity of inequality: if ''a'' < ''b'' and ''b'' < ''c'', then ''a'' < ''c''.
* If ''a'' < ''b'' and ''a'', ''b'' > 0, then 1/''b'' < 1/''a''.
* 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 ≤ ''a''2 for all ''a'' in ''F''.
* Every non-trivial sum of squares is nonzero. Equivalently: [
Every subfield of an ordered field is also an ordered field (inheriting the induced ordering). The smallest subfield is ]isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the rationals
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 rationa ...
(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 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 ...
and contains infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
s. For example, the real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s form an Archimedean field, but hyperreal numbers
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
form a non-Archimedean field, because it extends real numbers with elements greater than any standard natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
.
An ordered field ''F'' is isomorphic to the real number field R if every non-empty subset of ''F'' with an upper bound in ''F'' has a least upper bound
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
in ''F''. This property implies that the field is Archimedean.
Vector spaces over an ordered field
Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s (particularly, ''n''-spaces) over an ordered field exhibit some special properties and have some specific structures, namely: orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
, convexity
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
, and positively-definite inner product. See Real coordinate space#Geometric properties and uses for discussion of those properties of R''n'', which can be generalized to vector spaces over other ordered fields.
Orderability of fields
Every ordered field is a formally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field.
Alternative definitions
The definition given above is ...
, i.e., 0 cannot be written as a sum of nonzero squares.[Lam (2005) p. 41][Lam (2005) p. 232]
Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order need not be uniquely determined.) The proof uses Zorn's lemma.[Lam (2005) p. 236]
Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s and more generally fields of positive characteristic cannot be turned into ordered fields, because in characteristic ''p'', the element −1 can be written as a sum of (''p'' − 1) squares 12. The complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s also cannot be turned into an ordered field, as −1 is a square of the imaginary unit ''i''. Also, the ''p''-adic numbers cannot be ordered, since according to Hensel's lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to ...
Q2 contains a square root of −7, thus 12+12+12+22+()2=0, and Q''p'' (''p'' > 2) contains a square root of 1−''p'', thus (''p''−1)⋅12+()2=0.[The squares of the square roots and are in Q, but are <0, so that these roots cannot be in Q which means that their expansions are not periodic.]
Topology induced by the order
If ''F'' is equipped with the order 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.
If ''X'' is a totally ordered set, t ...
arising from the total order ≤, then the axioms guarantee that the operations + and × are continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
, so that ''F'' is a topological field
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is w ...
.
Harrison topology
The Harrison topology is a topology on the set of orderings ''X''''F'' of a formally real field ''F''. Each order can be regarded as a multiplicative group homomorphism from ''F''∗ onto ±1. Giving ±1 the discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
and ±1''F'' the product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
induces the subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
on ''X''''F''. The Harrison sets form a subbasis
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
for the Harrison topology. The product is a Boolean space
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first h ...
(compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, Hausdorff and totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
), and ''X''''F'' is a closed subset, hence again Boolean.[Lam (2005) p. 271][Lam (1983) pp. 1–2]
Fans and superordered fields
A fan on ''F'' is a preordering ''T'' with the property that if ''S'' is a subgroup of index 2 in ''F''∗ containing ''T'' − and not containing −1 then ''S'' is an ordering (that is, ''S'' is closed under addition).[Lam (1983) p. 39] A superordered field is a totally real field in which the set of sums of squares forms a fan.[Lam (1983) p. 45]
See also
*
*
*
*
*
*
*
*
*
Notes
References
*
*
*
{{DEFAULTSORT:Ordered Field
Real algebraic geometry
Ordered algebraic structures
Ordered groups