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 ${\displaystyle \leq }$ 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.

Examples of ordered fields are:

• the rational numbers
• the real numbers
• any subfield of an ordered field, such as the real algebraic numbers or computable numbers
• the field ${\displaystyle \mathbb {Q} (x)}$ of rational functions ${\displaystyle p(x)/q(x)}$, where ${\displaystyle p(x)}$ and ${\displaystyle q(x)}$ are polynomials with rational coefficients, ${\displaystyle q(x)\neq 0}$, can be made into an ordered field by fixing a real transcendental number ${\displaystyle \alpha }$ and defining ${\displaystyle p(x)/q(x)>0}$ if and only if ${\displaystyle p(\alpha )/q(\alpha )>0}$. This is equivalent to embedding ${\displaystyle \mathbb {Q} (x)}$ into ${\displaystyle \mathbb {R} }$ and restricting the ordering of ${\displaystyle \mathbb {R} }$ to an ordering of the image of ${\displaystyle \mathbb {Q} (x)}$.
• the field ${\displaystyle \mathbb {R} (x)}$ of rational functions ${\displaystyle p(x)/q(x)}$, where ${\displaystyle p(x)}$ and ${\displaystyle q(x)}$ are polynomials with real coefficients, ${\displaystyle q(x)\neq 0}$, can be made into an ordered field where the polynomial ${\displaystyle p(x)=x}$ is greater than any constant polynomial, by defining ${\displaystyle p(x)/q(x)>0}$ to mean that ${\displaystyle p_{n}/q_{m}>0}$, where ${\displaystyle p_{n}\neq 0}$ and ${\displaystyle q_{m}\neq 0}$ are the leading coefficients of ${\displaystyle p(x)=p_{n}x^{n}+\dots +p_{0}}$ and ${\displaystyle q(x)=q_{m}x^{m}+\dots +q_{0}}$, respectively. This ordered field is not Archimedean.
• The field ${\displaystyle \mathbb {R} ((x))}$ 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.

The property ${\displaystyle a>0\land x<y\Rightarrow ax<ay}$

The property ${\displaystyle x<y\Rightarrow a+x<a+y}$

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.
• Transitivity of inequality: if a < b and b < c, then a < c.
• 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 ${\displaystyle 1=1^{2}}$, it follows that ${\displaystyle 1\geq 0}$. Since ${\displaystyle 1\neq 0}$, we conclude ${\displaystyle 1>0}$.
• An ordered field has characteristic 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc. If the field had characteristic p > 0, then −1 would be the sum of p − 1 ones, but −1 is not positive.) In particular, finite fields cannot be ordered.
• Every non-trivial sum of squares is nonzero. Equivalently: ${\displaystyle \textstyle \sum _{k=1}^{n}a_{k}^{2}=0\;\Longrightarrow \;\forall k\;\colon a_{k}=0.}$[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. 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 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, because in characteristic p, the element −1 can be written as a sum of (p − 1) squares 1². 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 Q_p (p > 2) contains a square root of 1 − p, thus (p − 1)⋅1² + (√1 − p)² = 0.[6]

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 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 ${\displaystyle H(a)=\{P\in X_{F}:a\in P\}}$ 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]

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]

• Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb
• Ordered group – Group with a compatible partial orderPages displaying short descriptions of redirect targets
• Ordered ring – ring with a compatible total orderPages displaying wikidata descriptions as a fallback
• Ordered topological vector space
• Ordered vector space – Vector space with a partial order
• Partially ordered ring – Ring with a compatible partial order
• Partially ordered space – Partially ordered topological space
• Preorder field – Algebraic concept in measure theory, also referred to as an algebra of setsPages displaying short descriptions of redirect targets
• Riesz space – Partially ordered vector space, ordered as a lattice

• Lam, T. Y. (1983), Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, vol. 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. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023.
• Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001