Math 2.0

statement proposition

The minimal polynomial divides every polynomial vanishing at the element

Statement

Let L/KL/K be a field extension, let a∈La \in L be algebraic over KK, and let mam_a be its minimal polynomial. For every f∈K[x]f \in K[x] with f(a)=0f(a) = 0, the polynomial mam_a divides ff in K[x]K[x].

Remarks

The proof: polynomial division gives f=qma+rf = q m_a + r with deg⁡r<deg⁡ma\deg r < \deg m_a. Then r(a)=0r(a) = 0, so r=0r = 0 by the minimality of deg⁡ma\deg m_a. Equivalently, the kernel of the evaluation map K[x]→LK[x] \to L, f↦f(a)f \mapsto f(a), is the ideal (ma)(m_a). If aa is a root of ff and ff is monic and irreducible over KK, then f=maf = m_a.

Explanations

L1 human

Every polynomial equation that the element satisfies comes from the minimal polynomial. It is a multiple of the minimal polynomial.

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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 ringdefinitionDivisibility in a commutative ring, definition, no evidence levelDivisibility in a com…definitionDegree 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 extensiondefinitionDivision algorithm for polynomials, statement, no evidence levelDivision algorithm fo…statementAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionMinimal polynomial of an algebraic element, definition, no evidence levelMinimal polynomial of…definitionThe minimal polynomial divides every polynomial vanishing at the element, statement, no evidence levelThe minimal polynomia…statement
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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. Divisibility in a commutative ring definition
  8. Degree of a polynomial definition
  9. Polynomial evaluation definition
  10. Field definition
  11. Leading coefficient definition
  12. Root of a polynomial definition
  13. Subfield definition
  14. Monic polynomial definition
  15. Field extension definition
  16. Division algorithm for polynomials statement
  17. Algebraic element definition
  18. Minimal polynomial of an algebraic element definition
  19. The minimal polynomial divides every polynomial vanishing at the element statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Minimal_polynomial_(field_theory), Properties

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