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Algebraic object with an ordered structure
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 (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.

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 maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.

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.

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.

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.

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 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.

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.