Math 2.0

Field theory and polynomials Definition

Real closure

Builds on Ordered fieldFormally real fieldAlgebraic extensionAlgebraic number

Definition

A field RR is real closed if RR is formally real and no algebraic extension R′≠RR' \ne R of RR is formally real. Let (F,≤)(F, \le) be an ordered field. A real closure of (F,≤)(F, \le) is an algebraic extension RR of FF such that:

  1. RR is real closed;
  2. the ordering of RR, in which the nonnegative elements are the squares, restricts to ≤\le on FF.

Remarks

A real closed field has exactly one ordering, given by a≥0a \ge 0 if and only if aa is a square, so clause 2 makes sense. Equivalently, RR is real closed if and only if two things hold. It has an ordering in which every positive element is a square, and every polynomial of odd degree over RR has a root in RR. Every ordered field has a real closure, and it is unique up to a unique isomorphism over FF that respects the orderings. Example: the field of real algebraic numbers is a real closure of ℚ\Q. Non-example: ℝ\R is real closed, but it is not a real closure of ℚ\Q, because it is not algebraic over ℚ\Q.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

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Algebraic number, definition, no evidence levelAlgebraic numberdefinitionFormally real field, concept, no evidence levelFormally real fieldconceptGroup, concept, no evidence levelGroupconceptOrdered field, definition, no evidence levelOrdered fielddefinitionRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionField extension, definition, no evidence levelField extensiondefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinitionReal closure, definition, no evidence levelReal closuredefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic number definition
  2. Formally real field concept
  3. Group concept
  4. Ordered field definition
  5. Ring definition
  6. Commutative ring definition
  7. Polynomial ring definition
  8. Subring definition
  9. Unit of a ring definition
  10. Polynomial evaluation definition
  11. Field definition
  12. Root of a polynomial definition
  13. Subfield definition
  14. Field extension definition
  15. Algebraic element definition
  16. Algebraic extension definition
  17. Real closure definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Real_closed_field
About this page
TagKWDJF4JEKN
URIm2://KWDJF4JEKN
Kinddefinition
WritingW0 stub
MSC12J15, 12D15
DefinesReal closure
UsesOrdered field, Formally real field, Algebraic extension, Algebraic number
Changed2 November 2026

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