Math 2.0

Field theory and polynomials Field extensions Proposition

The minimal polynomial is irreducible

Builds on Minimal polynomial of an algebraic elementIrreducible polynomial

Statement

Let L/KL/K be a field extension and let a∈La \in L be algebraic over KK. Then the minimal polynomial mam_a of aa over KK is irreducible over KK.

Remarks

The proof: if ma=ghm_a = g h with g,h∈K[x]g, h \in K[x] of smaller degree, then g(a)h(a)=0g(a) h(a) = 0 in the field LL. So g(a)=0g(a) = 0 or h(a)=0h(a) = 0, and this contradicts the minimality of deg⁡ma\deg m_a. Conversely, a monic irreducible polynomial f∈K[x]f \in K[x] with f(a)=0f(a) = 0 is the minimal polynomial of aa. For example, x3−2x^3 - 2 is irreducible over ℚ\mathbb{Q}, so it is the minimal polynomial of 23\sqrt[3]{2} over ℚ\mathbb{Q}.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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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 divisordefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionMinimal polynomial of an algebraic element, definition, no evidence levelMinimal polynomial of…definitionThe minimal polynomial is irreducible, statement, no evidence levelThe minimal polynomia…statement
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.
  • no evidence level
  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. Degree of a polynomial definition
  9. Polynomial evaluation definition
  10. Field definition
  11. Integral domain definition
  12. Leading coefficient definition
  13. Root of a polynomial definition
  14. Subfield definition
  15. Irreducible element definition
  16. Monic polynomial definition
  17. Field extension definition
  18. Irreducible polynomial definition
  19. Algebraic element definition
  20. Minimal polynomial of an algebraic element definition
  21. The minimal polynomial is irreducible statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Minimal_polynomial_(field_theory), Irreducibility
About this page
Tag0NFEG48RDR
URIm2://0NFEG48RDR
Kindstatement, proposition
WritingW0 stub
MSC12F05
StatusThis copy of the site has no build data. Run m2 status 0NFEG48RDR.
UsesMinimal polynomial of an algebraic element, Irreducible polynomial

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