Math 2.0

Field theory and polynomials Definition

Algebraically closed field

Builds on FieldRoot of a polynomialSplit polynomialIrreducible polynomial

Definition

A field KK is algebraically closed if every polynomial f∈K[x]f \in K[x] with deg⁡f≥1\deg f \ge 1 has a root in KK.

Remarks

Equivalently, every nonzero polynomial over KK splits over KK. Also equivalently, the irreducible polynomials over KK are exactly the polynomials of degree one. Another equivalent condition: KK has no algebraic extension other than KK itself. The field ℂ\mathbb{C} is algebraically closed, by the fundamental theorem of algebra. The field ℝ\mathbb{R} is not, because x2+1x^2 + 1 has no real root. No finite field is algebraically closed: for a finite field KK, the polynomial 1+∏a∈K(x−a)1 + \prod_{a \in K} (x - a) has no root in 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 ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionField, definition, no evidence levelFielddefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionAlgebraically closed field, definition, no evidence levelAlgebraically closed …definition
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. Unit of a ring definition
  6. Zero divisor definition
  7. Field definition
  8. Polynomial evaluation definition
  9. Integral domain definition
  10. Root of a polynomial definition
  11. Irreducible element definition
  12. Split polynomial definition
  13. Irreducible polynomial definition
  14. Algebraically closed field definition

Where this leads

25 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Algebraically_closed_field, Equivalent properties
About this page
Tag02SJG1HC32
URIm2://02SJG1HC32
Kinddefinition
WritingW1 stated
MSC12F05
DefinesAlgebraically closed field
UsesField, Root of a polynomial, Split polynomial, Irreducible polynomial

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