Math 2.0

Field theory and polynomials Irreducibility Theorem

Reduction modulo p test for irreducibility

Builds on Irreducible polynomialMonic polynomialGauss's lemma

Statement

Let f∈ℤ[x]f \in \mathbb{Z}[x] be a monic polynomial and let pp be a prime number. Let fˉ∈𝔽p[x]\bar{f} \in \mathbb{F}_p[x] be the polynomial whose coefficients are those of ff modulo pp. If fˉ\bar{f} is irreducible in 𝔽p[x]\mathbb{F}_p[x], then ff is irreducible in ℤ[x]\mathbb{Z}[x] and in ℚ[x]\mathbb{Q}[x].

Remarks

The proof: a factorisation f=ghf = g h into monic factors of positive degree gives fˉ=gˉhˉ\bar{f} = \bar{g} \bar{h}, with deg⁡gˉ=deg⁡g\deg \bar{g} = \deg g and deg⁡hˉ=deg⁡h\deg \bar{h} = \deg h. Irreducibility in ℚ[x]\mathbb{Q}[x] then follows by Gauss's lemma. The converse is false: x4+1x^4 + 1 is irreducible over ℚ\mathbb{Q}, but it is reducible modulo every prime. Example: x3+x+1x^3 + x + 1 has no root in 𝔽2\mathbb{F}_2, so it is irreducible over 𝔽2\mathbb{F}_2, and so over ℚ\mathbb{Q}.

Reading path

To understand this page from the ground up, read these 24 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 ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionDivisibility in a commutative ring, definition, no evidence levelDivisibility in a com…definitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionField, definition, no evidence levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionAssociate elements, definition, no evidence levelAssociate elementsdefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionPrime element, definition, no evidence levelPrime elementdefinitionField of fractions, definition, no evidence levelField of fractionsdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionGreatest common divisor in an integral domain, definition, no evidence levelGreatest common divis…definitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionUnique factorisation domain, definition, no evidence levelUnique factorisation …definitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionContent of a polynomial, definition, no evidence levelContent of a polynomi…definitionPrimitive polynomial, definition, no evidence levelPrimitive polynomialdefinitionProduct of primitive polynomials is primitive, statement, no evidence levelProduct of primitive …statementGauss's lemma, statement, no evidence levelGauss's lemmastatementReduction modulo p test for irreducibility, statement, no evidence levelReduction modulo p te…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. Unit of a ring definition
  6. Zero divisor definition
  7. Divisibility in a commutative ring definition
  8. Degree of a polynomial definition
  9. Field definition
  10. Integral domain definition
  11. Associate elements definition
  12. Leading coefficient definition
  13. Prime element definition
  14. Field of fractions definition
  15. Irreducible element definition
  16. Greatest common divisor in an integral domain definition
  17. Monic polynomial definition
  18. Unique factorisation domain definition
  19. Irreducible polynomial definition
  20. Content of a polynomial definition
  21. Primitive polynomial definition
  22. Product of primitive polynomials is primitive statement
  23. Gauss's lemma statement
  24. Reduction modulo p test for irreducibility statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Irreducible_polynomial, Over the integers and finite fields
About this page
Tag0KWWEXW0RN
URIm2://0KWWEXW0RN
Kindstatement, theorem
WritingW0 stub
MSC12E05
StatusThis copy of the site has no build data. Run m2 status 0KWWEXW0RN.
UsesIrreducible polynomial, Monic polynomial, Gauss's lemma

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