Formally real field: Difference between revisions
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==Definition== | ==Definition== | ||
A '''formally real field''' is a [[field]] in which <math>-1</math> is not a | ===Algebraic definition=== | ||
A '''formally real field''' is a [[field]] in which <math>-1</math> is not a sum of squares. | |||
===Definition in terms of total ordering=== | |||
A formally real field is a [[field]] for which there exists a total ordering <math>\le</math> under which the field becomes an [[ordered field]]. In other words, <math>\le</math> satisfies the following conditions: | |||
* <math>a \le b</math> and <math>c \le d</math> implies <math>a + c \le b + d</math>. | |||
* <math>0 \le 1</math>. | |||
* <math>a \le b</math> implies <math>-b \le -a</math>. | |||
* <math>0 \le a</math> and <math>b \le c</math> implies <math>ab \le ac</math>. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Weaker than::Real-closed field]] | * [[Weaker than::Real-closed field]] | ||
* [[Weaker than::Archimedean field]] | |||
* [[Weaker than::Euclidean field]] | |||
===Weaker properties=== | |||
* [[Stronger than::Field of characteristic zero]]: {{proofofstrictimplicationat|[[Formally real implies characteristic zero]]|[[Characteristic zero not implies formally real]]}} | |||
Revision as of 22:04, 14 May 2009
This article defines a field property: a property that can be evaluated to true/false for any field.
View a complete list of field properties|View a complete list of field extension properties
Definition
Algebraic definition
A formally real field is a field in which is not a sum of squares.
Definition in terms of total ordering
A formally real field is a field for which there exists a total ordering under which the field becomes an ordered field. In other words, satisfies the following conditions:
- and implies .
- .
- implies .
- and implies .
Relation with other properties
Stronger properties
Weaker properties
- Field of characteristic zero: For proof of the implication, refer Formally real implies characteristic zero and for proof of its strictness (i.e. the reverse implication being false) refer Characteristic zero not implies formally real.