ordered field

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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 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 ( reflexiv ...
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 re ...
, and every Dedekind-complete 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 i ...
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 ratio ...
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 ad ...
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, subt ...
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 ...
,
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 of ordered fields and formally real fields.

# 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 $\,\leq\,$ 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 $\left(F, +, \cdot\,\right)$ together with a (strict) total order $\,<\,$ on $F$ is an if the order satisfies the following properties for all $a, b, c \in F:$ * if $a < b$ then $a + c < b + c,$ and * if $0 < a$ and $0 < b$ then $0 < a \cdot b.$

## Positive cone

A or preordering of a field $F$ is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset o ...
$P \subseteq F$ that has the following properties:Lam (2005) p. 289 * For $x$ and $y$ in $P,$ both $x + y$ and $x \cdot y$ are in $P.$ * If $x \in F,$ then $x^2 \in P.$ In particular, $1^2 = 1 \in P.$ * The element $- 1$ is not in $P.$ A is a field equipped with a preordering $P.$ Its non-zero elements $P^*$ 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 $F.$ If in addition, the set $F$ is the union of $P$ and $- P,$ we call $P$ a positive cone of $F.$ The non-zero elements of $P$ are called the positive elements of $F.$ An ordered field is a field $F$ together with a positive cone $P.$ The preorderings on $F$ are precisely the intersections of families of positive cones on $F.$ The positive cones are the maximal preorderings.

## 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 the first definition, the set of elements such that $x \geq 0$ forms a positive cone of $F.$ Conversely, given a positive cone $P$ of $F$ as in the second definition, one can associate a total ordering $\,\leq_P\,$ on $F$ by setting $x \leq_P y$ to mean $y - x \in P.$ This total ordering $\,\leq_P\,$ 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 ratio ...
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 ...
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 ...
or computable numbers * the field of real rational functions $\frac \,$, where $p\left(x\right)$ and $q\left(x\right)$ 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 examp ...
s with real coefficients, $q\left(x\right) \ne 0\,$, can be made into an ordered field where the polynomial $p\left(x\right)=x$ is greater than any constant polynomial, by defining that $\frac > 0\,$ whenever $\frac > 0\,$, for $p\left(x\right) = p_0\sdot x^n + \cdots$ and $q\left(x\right) = q_0\sdot x^m + \cdots\,$. This ordered field is not Archimedean. * The field $\mathbb\left(\left(x\right)\right)$ of formal Laurent series with real coefficients, where ''x'' is taken to be infinitesimal and positive * the transseries * real closed fields * the superreal numbers * 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 number ...
s The surreal numbers 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, 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: $\textstyle \sum_^n a_k^2 = 0 \; \Longrightarrow \; \forall k \; \colon a_k = 0 .$ 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 i ...
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 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 refer ...
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 ...
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 number ...
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 ...
. 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 ''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 ca ...
s (particularly, ''n''-spaces) over an ordered field exhibit some special properties and have some specific structures, namely: orientation, convexity, 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, i.e., 0 cannot be written as a sum of nonzero squares.Lam (2005) p. 41Lam (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, subt ...
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 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 t ...
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-seemi ...
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 $H\left(a\right) = \$ form a subbasis for the Harrison topology. The product is a Boolean space (
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 Briti ...
, 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. 271Lam (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