Math 2.0

Field theory and polynomials Definition

Minimal polynomial of an algebraic element

Builds on Algebraic elementMonic polynomialDegree of a polynomial

Definition

Let L/KL/K be a field extension and let a∈La \in L be algebraic over KK. The minimal polynomial of aa over KK is the monic polynomial ma∈K[x]m_a \in K[x] of smallest degree such that ma(a)=0m_a(a) = 0.

Remarks

The minimal polynomial exists and is unique. If two monic polynomials of the smallest degree vanish at aa, their difference has smaller degree and vanishes at aa, so it is zero. The minimal polynomial is irreducible, and it divides each polynomial that vanishes at aa. The minimal polynomial of 2\sqrt{2} over ℚ\mathbb{Q} is x2−2x^2 - 2. Over ℝ\mathbb{R}, it is x−2x - \sqrt{2}. The minimal polynomial of ii over ℝ\mathbb{R} is x2+1x^2 + 1.

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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 polynomialdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionRoot 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…definition
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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. Polynomial evaluation definition
  9. Field definition
  10. Leading coefficient definition
  11. Root of a polynomial definition
  12. Subfield definition
  13. Monic polynomial definition
  14. Field extension definition
  15. Algebraic element definition
  16. Minimal polynomial of an algebraic element definition

Where this leads

52 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Minimal_polynomial_(field_theory), Definition
About this page
Tag0GK1WB9R98
URIm2://0GK1WB9R98
Kinddefinition
WritingW1 stated
MSC12F05
DefinesMinimal polynomial (field theory)
UsesAlgebraic element, Monic polynomial, Degree of a polynomial

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