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In mathematics, an 'ordered field' is a field together with an ordering of its elements. This concept was introduced by Emil Artin in 1927.
There are two equivalent definitions, depending on which properties one takes as the definition for an ordered field.
A field (''F'',+,
★ ) together with a total order ≤ on ''F'' is an 'ordered field' if the order satisfies the following properties:
★ if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''
★ if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''a b''
It follows from these axioms that for every ''a'', ''b'', ''c'', ''d'' in ''F'':
★ Either −''a'' ≤ 0 ≤ ''a'' or ''a'' ≤ 0 ≤ −''a''.
★ We are allowed to "add inequalities": If ''a'' ≤ ''b'' and ''c'' ≤ ''d'', then ''a'' + ''c'' ≤ ''b'' + ''d''
★ We are allowed to "multiply inequalities with positive elements": If ''a'' ≤ ''b'' and 0 ≤ ''c'', then ''ac'' ≤ ''bc''.
An 'ordering' of a field ''F'' is a subset ''P'' ⊂ ''F'' that has the following properties:
★ ''F'' is the disjoint union of ''P'', −''P'', and the element 0. That is, for each ''x'' ∈ ''F'', then exactly one of the following conditions is true: ''x'' = 0, ''x'' ∈ ''P'' or −''x'' ∈ ''P''.
★ For ''x'' and ''y'' in ''P'', both ''x''+''y'' and ''xy'' are in ''P''.
The subset ''P'' are called the 'positive' elements of ''F''.
We next define ''x'' < ''y'' to mean that ''y'' − ''x'' ∈ ''P'' (so that ''y'' − ''x'' > 0 in a sense). This relation satisfies the expected properties:
★ If ''x'' < ''y'' and ''y'' < ''z'', then ''x'' < ''z''. (transitivity)
★ If ''x'' < ''y'' and ''z'' > 0, then ''xz'' < ''yz''.
★ If ''x'' < ''y'' and ''x'',''y'' > 0, then 1/''y'' < 1/''x''
The statement ''x'' ≤ ''y'' will mean that either ''x'' < ''y'' or ''x'' = ''y''.
★ 1 is positive. (Justification: either 1 is positive or −1 is positive. If −1 is positive, then (−1)(−1) is positive, which is a contradiction)
★ 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.
★ Squares are non-negative. 0 ≤ ''a''² for all ''a'' in ''F''. (Follows by a similar argument to 1 > 0)
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''. For example, the real numbers form an Archimedean field, but every hyperreal field is non-Archimedean.
An ordered field K is the real number field if it satisfies the axiom of Archimedes and the Cauchy sequence of K converges within K.
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.
Examples of ordered fields are:
★ the rational numbers
★ the real algebraic numbers
★ the computable numbers
★ the real numbers
★ the field of real rational functions , where p(x) and q(x), are polynomials with real coefficients, can be made into an ordered field where the polynomial p(x) = x is greater than any constant polynomial, by defining that whenever , for . This ordered field is not Archimedean.
★ The field of formal Laurent series with real coefficients , where x is taken to be infinitesimal and positive
★ real closed fields
★ superreal numbers
★ 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.
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 is often not uniquely determined.)
Finite fields cannot be turned into ordered fields, because they do not have characteristic 0. The complex numbers also cannot be turned into an ordered field, as −1 is a square (of the imaginary number ''i'') and would thus be positive. Also, the p-adic numbers cannot be ordered, since 'Q'2 contains a square root of −7 and 'Q'p (''p'' > 2) contains a square root of 1 − ''p''.
★ Algebra, Lang, Serge, , , Addison-Wesley, 1997, ISBN 978-0-201-55540-0
| Contents |
| Definition |
| Def 1: A total order on ''F'' |
| Def 2: An ''ordering'' on ''F'' |
| Properties of ordered fields |
| Topology induced by the order |
| Examples of ordered fields |
| Which fields can be ordered? |
| References |
Definition
There are two equivalent definitions, depending on which properties one takes as the definition for an ordered field.
Def 1: A total order on ''F''
A field (''F'',+,
★ ) together with a total order ≤ on ''F'' is an 'ordered field' if the order satisfies the following properties:
★ if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''
★ if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''a b''
It follows from these axioms that for every ''a'', ''b'', ''c'', ''d'' in ''F'':
★ Either −''a'' ≤ 0 ≤ ''a'' or ''a'' ≤ 0 ≤ −''a''.
★ We are allowed to "add inequalities": If ''a'' ≤ ''b'' and ''c'' ≤ ''d'', then ''a'' + ''c'' ≤ ''b'' + ''d''
★ We are allowed to "multiply inequalities with positive elements": If ''a'' ≤ ''b'' and 0 ≤ ''c'', then ''ac'' ≤ ''bc''.
Def 2: An ''ordering'' on ''F''
An 'ordering' of a field ''F'' is a subset ''P'' ⊂ ''F'' that has the following properties:
★ ''F'' is the disjoint union of ''P'', −''P'', and the element 0. That is, for each ''x'' ∈ ''F'', then exactly one of the following conditions is true: ''x'' = 0, ''x'' ∈ ''P'' or −''x'' ∈ ''P''.
★ For ''x'' and ''y'' in ''P'', both ''x''+''y'' and ''xy'' are in ''P''.
The subset ''P'' are called the 'positive' elements of ''F''.
We next define ''x'' < ''y'' to mean that ''y'' − ''x'' ∈ ''P'' (so that ''y'' − ''x'' > 0 in a sense). This relation satisfies the expected properties:
★ If ''x'' < ''y'' and ''y'' < ''z'', then ''x'' < ''z''. (transitivity)
★ If ''x'' < ''y'' and ''z'' > 0, then ''xz'' < ''yz''.
★ If ''x'' < ''y'' and ''x'',''y'' > 0, then 1/''y'' < 1/''x''
The statement ''x'' ≤ ''y'' will mean that either ''x'' < ''y'' or ''x'' = ''y''.
Properties of ordered fields
★ 1 is positive. (Justification: either 1 is positive or −1 is positive. If −1 is positive, then (−1)(−1) is positive, which is a contradiction)
★ 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.
★ Squares are non-negative. 0 ≤ ''a''² for all ''a'' in ''F''. (Follows by a similar argument to 1 > 0)
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''. For example, the real numbers form an Archimedean field, but every hyperreal field is non-Archimedean.
An ordered field K is the real number field if it satisfies the axiom of Archimedes and the Cauchy sequence of K converges within K.
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.
Examples of ordered fields
Examples of ordered fields are:
★ the rational numbers
★ the real algebraic numbers
★ the computable numbers
★ the real numbers
★ the field of real rational functions , where p(x) and q(x), are polynomials with real coefficients, can be made into an ordered field where the polynomial p(x) = x is greater than any constant polynomial, by defining that whenever , for . This ordered field is not Archimedean.
★ The field of formal Laurent series with real coefficients , where x is taken to be infinitesimal and positive
★ real closed fields
★ superreal numbers
★ 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.
Which fields can be ordered?
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 is often not uniquely determined.)
Finite fields cannot be turned into ordered fields, because they do not have characteristic 0. The complex numbers also cannot be turned into an ordered field, as −1 is a square (of the imaginary number ''i'') and would thus be positive. Also, the p-adic numbers cannot be ordered, since 'Q'2 contains a square root of −7 and 'Q'p (''p'' > 2) contains a square root of 1 − ''p''.
References
★ Algebra, Lang, Serge, , , Addison-Wesley, 1997, ISBN 978-0-201-55540-0
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