Ordered field: Difference between revisions
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In [[mathematics]], an '''ordered field''' is a [[field (mathematics)|field]] which has a [[linear order]] structure which is compatible with the field operations. | In [[mathematics]], an '''ordered field''' is a [[field (mathematics)|field]] which has a [[linear order]] structure which is compatible with the field operations. | ||
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It is possible for a field to have more than one linear order compatible with the field operations, but in any case the squares must lie in the positive cone. | It is possible for a field to have more than one linear order compatible with the field operations, but in any case the squares must lie in the positive cone. | ||
A field ''F'' | ==Artin-Schreier theorem== | ||
A field ''F'' is '''formally real''' if -1 is not a sum of squares in ''F''. The Artin-Schreier theorem states that a field ''F'' can be ordered if and only if it is formally real. | |||
==Examples== | ==Examples== | ||
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* The complex numbers cannot be given an ordered field structure since both 1 and -1 are squares. | * The complex numbers cannot be given an ordered field structure since both 1 and -1 are squares. | ||
* The [[quadratic field]] <math>\mathbf{Q}(\sqrt 2)</math> has two possible structures as ordered field, corresponding to the [[embedding]]s into '''R''' in which <math>\sqrt 2</math> takes on the two possible real values. | * The [[quadratic field]] <math>\mathbf{Q}(\sqrt 2)</math> has two possible structures as ordered field, corresponding to the [[embedding]]s into '''R''' in which <math>\sqrt 2</math> takes on the two possible real values. | ||
* No [[finite field]] can be ordered. | * No [[finite field]] can be ordered.[[Category:Suggestion Bot Tag]] |
Latest revision as of 12:01, 29 September 2024
In mathematics, an ordered field is a field which has a linear order structure which is compatible with the field operations.
Formally, F is an ordered field if there is a linear order ≤ on F which satisfies
- If then
- For each element or ;
- If and then
Alternatively, the order may be defined in terms of a positive cone, a subset C of F which is closed under addition and multiplication, contains the 0 and 1 elements, and which has the properties that
The relationship between the order and the associated positive cone is that
It is possible for a field to have more than one linear order compatible with the field operations, but in any case the squares must lie in the positive cone.
Artin-Schreier theorem
A field F is formally real if -1 is not a sum of squares in F. The Artin-Schreier theorem states that a field F can be ordered if and only if it is formally real.
Examples
- The rational numbers form an ordered field in a unique way.
- The real numbers form an ordered field in a unique way: the squares form the positive cone.
- The complex numbers cannot be given an ordered field structure since both 1 and -1 are squares.
- The quadratic field has two possible structures as ordered field, corresponding to the embeddings into R in which takes on the two possible real values.
- No finite field can be ordered.