Math 2.0

Field theory and polynomials Definition

Separable extension

Builds on Algebraic extensionMinimal polynomial of an algebraic elementSeparable polynomial

Definition

Let L/KL/K be an algebraic extension. An element a∈La \in L is separable over KK if its minimal polynomial over KK is a separable polynomial. The extension L/KL/K is separable if every element of LL is separable over KK.

Remarks

An irreducible polynomial over KK is separable if and only if its derivative is not the zero polynomial. So every algebraic extension of a field of characteristic 00 is separable. Every algebraic extension of a finite field is separable. For a prime pp and the field K=𝔽p(t)K = \mathbb{F}_p(t) of rational functions, the extension K(t1/p)/KK(t^{1/p})/K is not separable: the minimal polynomial xp−tx^p - t equals (x−t1/p)p(x - t^{1/p})^p.

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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 ringdefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionFormal derivative, definition, no evidence levelFormal derivativedefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionSeparable polynomial, definition, no evidence levelSeparable polynomialdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionMinimal polynomial of an algebraic element, definition, no evidence levelMinimal polynomial of…definitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinitionSeparable extension, definition, no evidence levelSeparable extensiondefinition
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. Degree of a polynomial definition
  8. Formal derivative definition
  9. Polynomial evaluation definition
  10. Field definition
  11. Leading coefficient definition
  12. Separable polynomial definition
  13. Root of a polynomial definition
  14. Subfield definition
  15. Monic polynomial definition
  16. Field extension definition
  17. Algebraic element definition
  18. Minimal polynomial of an algebraic element definition
  19. Algebraic extension definition
  20. Separable extension definition

Where this leads

23 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Separable_extension, Separable elements and separable extensions
About this page
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Kinddefinition
WritingW1 stated
MSC12F10
DefinesSeparable extension
UsesAlgebraic extension, Minimal polynomial of an algebraic element, Separable polynomial

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