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An ordered field with no proper algebraic ordered extension, equivalently whose algebraic closure is obtained by adjoining .
A real-closed field has a unique compatible ordering: its positive elements are exactly the nonzero squares.
wp-real-closednlab-real-closedmilne-algebraic-number-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require the ordered field to have no proper algebraic ordered extension.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-real-closednlab-real-closedmilne-algebraic-number-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
The real numbers are real closed; forgetting their completeness and topology retains that algebraic ordered-field property.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
nlab-real-numberEach relation below starts at this concept.
No direct relations are authored in this direction.