Math 2.0

Field theory and polynomials Definition

Algebraic closure

Builds on Algebraic extensionAlgebraically closed field

Definition

Let KK be a field. An algebraic closure of KK is a field extension K‾/K\overline{K}/K such that K‾/K\overline{K}/K is algebraic and K‾\overline{K} is algebraically closed.

Remarks

The field ℂ\mathbb{C} is an algebraic closure of ℝ\mathbb{R}, by the fundamental theorem of algebra. The field ℚ‾\overline{\mathbb{Q}} of algebraic numbers is an algebraic closure of ℚ\mathbb{Q}. The field ℂ\mathbb{C} is not, because ℂ/ℚ\mathbb{C}/\mathbb{Q} is not algebraic. If KK is infinite, then K‾\overline{K} has the same cardinality as KK. If KK is finite, then K‾\overline{K} is countably infinite. An algebraic closure is a largest algebraic extension of KK, and a smallest algebraically closed field that contains KK.

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Group, concept, no evidence levelGroupconceptRing, 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 ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionAlgebraically closed field, definition, no evidence levelAlgebraically closed …definitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinitionAlgebraic closure, definition, no evidence levelAlgebraic 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.
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Subring definition
  6. Unit of a ring definition
  7. Zero divisor definition
  8. Polynomial evaluation definition
  9. Field definition
  10. Integral domain definition
  11. Root of a polynomial definition
  12. Subfield definition
  13. Irreducible element definition
  14. Split polynomial definition
  15. Field extension definition
  16. Irreducible polynomial definition
  17. Algebraic element definition
  18. Algebraically closed field definition
  19. Algebraic extension definition
  20. Algebraic closure definition

Where this leads

12 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraic_closure, lead section
About this page
Tag0MCXX29HWY
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Kinddefinition
WritingW1 stated
MSC12F05
DefinesAlgebraic closure
UsesAlgebraic extension, Algebraically closed field

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