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 −1 (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 edit

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.

Total order edit

A field   together with a total order   on   is an ordered field if the order satisfies the following properties for all  

  • if   then   and
  • if   and   then  

As usual, we write   for   and  . The notations   and   stand for   and  , respectively. Elements   with   are called positive.

Positive cone edit

A prepositive cone or preordering of a field   is a subset   that has the following properties:[1]

  • For   and   in   both   and   are in  
  • If   then   In particular,   and  
  • The element   is not in  

A preordered field is a field equipped with a preordering   Its non-zero elements   form a subgroup of the multiplicative group of  

If in addition, the set   is the union of   and   we call   a positive cone of   The non-zero elements of   are called the positive elements of  

An ordered field is a field   together with a positive cone  

The preorderings on   are precisely the intersections of families of positive cones on   The positive cones are the maximal preorderings.[1]

Equivalence of the two definitions edit

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.

Examples of ordered fields edit

Examples of ordered fields are:

  • the field   of rational numbers with its standard ordering (which is also its only ordering);
  • the field   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   of rational 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   of rational 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   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 edit

The property  
The property  

For every a, b, c, d in F:

  • Either −a ≤ 0 ≤ a or a ≤ 0 ≤ −a.
  • One can "add inequalities": if ab and cd, then a + cb + d.
  • One can "multiply inequalities with positive elements": if ab and 0 ≤ c, then acbc.
  • "Multiplying with negatives flips an inequality": if ab and c ≤ 0, then acbc.
  • 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:  [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. 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 edit

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 edit

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 + −7)2 = 0, and Qp (p > 2) contains a square root of 1 − p, thus (p − 1)⋅12 + (1 − p)2 = 0.[6]

Topology induced by the order edit

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 edit

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.[7][8]

Fans and superordered fields edit

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]

See also edit

Notes edit

  1. ^ a b Lam (2005) p. 289
  2. ^ a b Lam (2005) p. 41
  3. ^ a b Lam (2005) p. 232
  4. ^ Bair, Jaques; Henry, Valérie. "Implicit differentiation with microscopes" (PDF). University of Liège. Retrieved 2013-05-04.
  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.
  7. ^ Lam (2005) p. 271
  8. ^ Lam (1983) pp. 1–2
  9. ^ Lam (1983) p. 39
  10. ^ Lam (1983) p. 45

References edit