Vipul: Created page with '==Definition== A '''real-closed field''' is a field satisfying the following equivalent conditions: # It is elementarily equivalent to the field of real numbers. # It is a ...'

2009-05-10T17:16:04Z

Created page with '==Definition== A '''real-closed field''' is a field satisfying the following equivalent conditions: # It is elementarily equivalent to the field of real numbers. # It is a ...'

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==Definition==

A '''real-closed field''' is a field satisfying the following equivalent conditions:

# It is elementarily equivalent to the [[field of real numbers]].
# It is a [[defining ingredient::Euclidean field]] (i.e., it is [[formally real field|formally real]] and for every element <math>D</math>, <math>D</math> or <math>-D</math> is a square) and every polynomial of odd degree over the field has a root.
# Its algebraic closure is a degree two extension of it, given as the splitting field of the polynomial <math>x^2 + 1</math>.

==Examples==

* [[Field of real numbers]]
* [[Field of real algebraic numbers]]

==Relation with other properties==

===Weaker properties===

* [[Stronger than::Euclidean field]]
* [[Stronger than::Pythagorean field]]
* [[Stronger than::Formally real field]]
* [[Stronger than::Field with trivial automorphism group]]

Real-closed field - Revision history

Vipul: Created page with '==Definition== A '''real-closed field''' is a field satisfying the following equivalent conditions: # It is elementarily equivalent to the field of real numbers. # It is a ...'