Inmathematics, 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 unitiis−1 (which is negative in any ordered field). Finite fields cannot be ordered.
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.
Let be a field. There is a bijection between the field orderings of and the positive cones of
Given a field ordering ≤ as in the first definition, the set of elements such that forms a positive cone of Conversely, given a positive cone of as in the second definition, one can associate a total ordering on by setting to mean This total ordering satisfies the properties of the first definition.
the field ofrational numbers with its standard ordering (which is also its only ordering);
the field ofreal 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 ofrational functions, where and are polynomials with rational coefficients and , can be made into an ordered field by fixing a real transcendental number and defining if and only if . This is equivalent to embedding into via and restricting the ordering of to an ordering of the image of . In this fashion, we get many different orderings of .
the field ofrational functions, where and are polynomials with real coefficients and , can be made into an ordered field by defining to mean that , where and are the leading coefficients of and , respectively. Equivalently: for rational functions we have if and only if for all sufficiently large . In this ordered field the polynomial is greater than any constant polynomial and the ordered field is not Archimedean.
The field offormal Laurent series with real coefficients, where x is taken to be infinitesimal and positive
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.
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.
Ifa < b and a, b > 0, then 1/b < 1/a.
Squares are non-negative: 0 ≤ a2 for all ainF. 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: [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.
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.
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 lemmaQ2 contains a square root of −7, thus 12 + 12 + 12 + 22 + √−72 = 0, and Qp (p > 2) contains a square root of 1 − p, thus (p − 1)⋅12 + √1 − p2 = 0.[6]
IfF 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 topologyonXF. 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.[7][8]
AfanonF 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]Asuperordered field is a totally real field in which the set of sums of squares forms a fan.[10]
^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.