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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...
and the
real numbers In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
, both with their standard orderings. 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 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 (for example, The set of all ...
s. Every Dedekind-complete ordered field is isomorphic to the reals.
Squares In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
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 for ...
s cannot be ordered since the square of the imaginary unit ''i'' is (which is negative in any ordered field).
Finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s cannot be ordered. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn. This grew eventually into the Artin–Schreier theory 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 ...
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 \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 (F, +, \cdot\,) together with a total order \leq on F is an if the order satisfies the following properties for all a, b, c \in F: * if a \leq b then a + c \leq b + c, and * if 0 \leq a and 0 \leq b then 0 \leq a \cdot b. As usual, we write a < b for a\le b and a\ne b. The notations b\ge a and b> a stand for a\le b and a < b, respectively. Elements a\in F with a>0 are called positive.


Positive cone

A or preordering of a field F is a
subset In mathematics, a 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 a ...
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, 0 = 0^2 \in P and 1 = 1^2 \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, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
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 field \Q of
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 (for example, The set of all ...
s with its standard ordering (which is also its only ordering); * the field \R of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s with its standard ordering (which is also its only ordering); * any subfield of an ordered field, such as the real algebraic numbers or the computable numbers, becomes an ordered field by restricting the ordering to the subfield; * the field \mathbb(x) of rational functions p(x)/q(x), where p(x) and q(x) are polynomials with rational coefficients and q(x) \ne 0, can be made into an ordered field by fixing a real transcendental number \alpha and defining p(x)/q(x) > 0 if and only if p(\alpha)/q(\alpha) > 0. This is equivalent to embedding \mathbb(x) into \mathbb via x\mapsto \alpha and restricting the ordering of \mathbb to an ordering of the image of \mathbb(x). In this fashion, we get many different orderings of \mathbb(x). * the field \mathbb(x) of rational functions p(x)/q(x), where p(x) and q(x) are polynomials with real coefficients and q(x) \ne 0, can be made into an ordered field by defining p(x)/q(x) > 0 to mean that p_n/q_m > 0, where p_n \neq 0 and q_m \neq 0 are the leading coefficients of p(x) = p_n x^n + \dots + p_0 and q(x) = q_m x^m + \dots + q_0, respectively. Equivalently: for rational functions f(x), g(x)\in \mathbb(x) we have f(x) < g(x) if and only if f(t) < g(t) for all sufficiently large t\in\mathbb. In this ordered field the polynomial p(x)=x is greater than any constant polynomial and the ordered field is not Archimedean. * The field \mathbb((x)) 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 numbers 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.


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''. * "Multiplying with negatives flips an inequality": if ''a'' ≤ ''b'' and c ≤ 0, then ''ac'' ≥ ''bc''. * If ''a'' < ''b'' and ''a'', ''b'' > 0, then 1/''b'' < 1/''a''. * Squares are non-negative: 0 ≤ ''a''2 for all ''a'' in ''F''. In particular, since 1=12, it follows that 0 ≤ 1. Since 0 ≠ 1, we conclude 0 < 1. * An ordered field has characteristic 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc., and no finite sum of ones can equal zero.) In particular, finite fields cannot be ordered. * 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 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 infinitesimals. 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s form an Archimedean field, but hyperreal 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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
. An ordered field ''F'' is isomorphic to the real number field R if and only 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 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 ...
, 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 Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
.Lam (2005) p. 236
Finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s and more generally fields of positive characteristic cannot be turned into ordered fields, as shown above. 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 for ...
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 arising from the total order ≤, then the axioms guarantee that the operations + and × are continuous, so that ''F'' is a topological field.


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 induces the
subspace topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
on ''X''''F''. The Harrison sets H(a) = \ form a subbasis for the Harrison topology. The product is a Boolean space ( compact, Hausdorff and totally disconnected), 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


See also

* * * * * * * * *


Notes


References

* * * {{DEFAULTSORT:Ordered Field Real algebraic geometry Ordered algebraic structures Ordered groups