Ordered field

In mathematics, 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 and the real numbers, 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 numbers. Every Dedekind-complete ordered field is isomorphic to the reals. Squares are necessarily non-negative in an ordered field. This implies that the complex numbers cannot be ordered since the square of the imaginary unit i is Template:Num (which is negative in any ordered field). Finite fields 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 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.

Total order

A field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (F, +, \cdot\,)} together with a total order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq } on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is an ordered field if the order satisfies the following properties for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in F:}

if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \leq b} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a + c \leq b + c,} and
if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq b} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \leq a \cdot b.}

As usual, we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a < b} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\le b } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\ne b} . The notations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\ge a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b> a} stand for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\le b} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a < b} , respectively. Elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in F} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a>0} are called positive.

Positive cone

A prepositive cone or preordering of a field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is a subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \subseteq F} that has the following properties:^[1]

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P,} both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \cdot y} are in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P.}
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in F,} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 \in P.} In particular, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 = 0^2 \in P} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 = 1^2 \in P.}
The element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - 1} is not in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P.}

A preordered field is a field equipped with a preordering Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P.} Its non-zero elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^*} form a subgroup of the multiplicative group of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F.}

If in addition, the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is the union of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - P,} we call Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} a positive cone of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F.} The non-zero elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} are called the positive elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F.}

An ordered field is a field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} together with a positive cone Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P.}

The preorderings on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} are precisely the intersections of families of positive cones on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F.} The positive cones are the maximal preorderings.^[1]

Equivalence of the two definitions

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} be a field. There is a bijection between the field orderings of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} and the positive cones of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F.}

Given a field ordering ≤ as in the first definition, the set of elements such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \geq 0} forms a positive cone of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F.} Conversely, given a positive cone Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} as in the second definition, one can associate a total ordering Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq_P} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} by setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \leq_P y} to mean Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - x \in P.} This total ordering Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \leq_P} satisfies the properties of the first definition.

Examples of ordered fields

Examples of ordered fields are:

the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Q} of rational numbers with its standard ordering (which is also its only ordering);
the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} of real numbers 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}(x)} of rational functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)/q(x)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q(x)} are polynomials with rational coefficients and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q(x) \ne 0} , can be made into an ordered field by fixing a real transcendental number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} and defining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)/q(x) > 0} if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(\alpha)/q(\alpha) > 0} . This is equivalent to embedding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}(x)} into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} via Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\mapsto \alpha} and restricting the ordering of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} to an ordering of the image of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}(x)} . In this fashion, we get many different orderings of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}(x)} .
the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}(x)} of rational functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)/q(x)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q(x)} are polynomials with real coefficients and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q(x) \ne 0} , can be made into an ordered field by defining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)/q(x) > 0} to mean that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n/q_m > 0} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n \neq 0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_m \neq 0} are the leading coefficients of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = p_n x^n + \dots + p_0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q(x) = q_m x^m + \dots + q_0} , respectively. Equivalently: for rational functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x), g(x)\in \mathbb{R}(x)} we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) < g(x)} if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t) < g(t)} for all sufficiently large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t\in\mathbb{R}} . In this ordered field the polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)=x} is greater than any constant polynomial and the ordered field is not Archimedean.
The field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}((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

File:Invariance of less-than-relation by multiplication with positive number.svg

The property Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a > 0 \land x < y \Rightarrow ax < ay}

File:Translation invariance of less-than-relation.svg

The property Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x < y \Rightarrow a+x < a+y}

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 ≤ a² 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \sum_{k=1}^n a_k^2 = 0 \; \Longrightarrow \; \forall k \; \colon a_k = 0 .} ^[2]^[3]

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 as well as infinite elements larger than any rational. For example, the real numbers 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.^[4]

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 spaces (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 Rⁿ, 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.^[2]^[3]

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.^[5]

Finite fields and more generally fields of positive characteristic cannot be turned into ordered fields, as shown above. The complex numbers 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 + √−7² = 0, and Q_p (p > 2) contains a square root of 1 − p, thus (p − 1)⋅1² + (√1 − p)² = 0.^[6]

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 and ±1^F the product topology induces the subspace topology on X_F. The Harrison sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(a) = \{ P \in X_F : a \in P \}} 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.^[7]^[8]

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 − {0} and not containing −1 then S is an ordering (that is, S is closed under addition).^[9] A superordered field is a totally real field in which the set of sums of squares forms a fan.^[10]

Notes

↑ ^1.0 ^1.1 Lam (2005) p. 289
↑ ^2.0 ^2.1 Lam (2005) p. 41
↑ ^3.0 ^3.1 Lam (2005) p. 232
↑ Bair, Jaques; Henry, Valérie. "Implicit differentiation with microscopes" (PDF). University of Liège. Retrieved 2013-05-04.
↑ Lam (2005) p. 236
↑ The squares of the square roots √−7 and √1 − p are in Q, but are < 0, so that these roots cannot be in Q which means that their p-adic expansions are not periodic.
↑ Lam (2005) p. 271
↑ Lam (1983) pp. 1–2
↑ Lam (1983) p. 39
↑ Lam (1983) p. 45

References

Lam, T. Y. (1983), Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, 52, American Mathematical Society, ISBN 0-8218-0702-1, Zbl 0516.12001
Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023.
Template:Lang Algebra

Template:Order theory

[Lam289-1] 1.0 ^1.1 Lam (2005) p. 289

[Lam41-2] 2.0 ^2.1 Lam (2005) p. 41

[Lam232-3] 3.0 ^3.1 Lam (2005) p. 232

[BairHenry-4] Bair, Jaques; Henry, Valérie. "Implicit differentiation with microscopes" (PDF). University of Liège. Retrieved 2013-05-04.

[Lam236-5] Lam (2005) p. 236

[6] The squares of the square roots √−7 and √1 − p are in Q, but are < 0, so that these roots cannot be in Q which means that their p-adic expansions are not periodic.

[Lam271-7] Lam (2005) p. 271

[L8312-8] Lam (1983) pp. 1–2

[L8339-9] Lam (1983) p. 39

[L8345-10] Lam (1983) p. 45

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Ordered field

Contents

Definitions

Total order

Positive cone

Equivalence of the two definitions

Examples of ordered fields

Properties of ordered fields

Vector spaces over an ordered field

Orderability of fields

Topology induced by the order

Harrison topology

Fans and superordered fields

See also

Notes

References

Navigation menu

Ordered field

Definitions

Total order

Positive cone

Equivalence of the two definitions

Examples of ordered fields

Properties of ordered fields

Vector spaces over an ordered field

Orderability of fields

Topology induced by the order

Harrison topology

Fans and superordered fields

See also

Notes

References

Navigation menu

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