Ordered field

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 for a\le b and a\ne b. The notations b\ge a and b> a stand for a\le b and a , respectively. Elements a\in F with a>0 are called positive. Positive cone A ' or preordering' of a field F is a subset P \subseteq F that has the following properties: • 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 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 numbers with its standard ordering (which is also its only ordering); • the field \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 \mathbb{Q}(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{Q}(x) into \mathbb{R} via x\mapsto \alpha and restricting the ordering of \mathbb{R} to an ordering of the image of \mathbb{Q}(x). In this fashion, we get many different orderings of \mathbb{Q}(x). • the field \mathbb{R}(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{R}(x) we have f(x) if and only if f(t) for all sufficiently large t\in\mathbb{R}. 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{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==

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 0, then 1/b 2 for all a in F. In particular, since 1=12, it follows that 0 ≤ 1. Since 0 ≠ 1, we conclude 0 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_{k=1}^n a_k^2 = 0 \; \Longrightarrow \; \forall k \; \colon a_k = 0 . 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. 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. 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 + 2 = 0, and Qp (p > 2) contains a square root of 1 − p, thus (p − 1)⋅12 + ()2 = 0. ==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 XF 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 ±1F the product topology induces the subspace topology on XF. The Harrison sets 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 XF is a closed subset, hence again Boolean. ==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). A superordered field is a totally real field in which the set of sums of squares forms a fan. == See also ==