Math 2.0

Field theory and polynomials Irreducibility Theorem

Eisenstein's criterion

Builds on Irreducible polynomialGauss's lemmaPrimitive polynomial

Statement

Let f=anxn+an−1xn−1+⋯+a0∈ℤ[x]f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in \mathbb{Z}[x] with n≥1n \ge 1, and let pp be a prime number with these three properties.

  1. pp divides aia_i for each ii with 0≤i<n0 \le i < n.
  2. pp does not divide ana_n.
  3. p2p^2 does not divide a0a_0.

Then ff is irreducible in ℚ[x]\mathbb{Q}[x].

Remarks

If ff is also primitive, then ff is irreducible in ℤ[x]\mathbb{Z}[x], by Gauss's lemma. For example, xn−2x^n - 2 is irreducible over ℚ\mathbb{Q} for each n≥1n \ge 1, with p=2p = 2. A change of variable can help: f(x+1)f(x + 1) is irreducible if and only if f(x)f(x) is. This proves that xp−1+⋯+x+1x^{p-1} + \dots + x + 1 is irreducible for pp prime. The criterion holds more generally for a prime ideal PP of an integral domain RR, with P2P^2 in place of p2p^2.

History

Theodor Schönemann published a version of the criterion in 1846. Gotthold Eisenstein published a different version in 1850. Both papers appeared in Crelle's Journal.

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 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…definitionField, definition, no evidence levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionAssociate elements, definition, no evidence levelAssociate elementsdefinitionPrime 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…definitionUnique 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 lemmastatementEisenstein's criterion, statement, no evidence levelEisenstein's criterionstatement
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. Field definition
  9. Integral domain definition
  10. Associate elements definition
  11. Prime element definition
  12. Field of fractions definition
  13. Irreducible element definition
  14. Greatest common divisor in an integral domain definition
  15. Unique factorisation domain definition
  16. Irreducible polynomial definition
  17. Content of a polynomial definition
  18. Primitive polynomial definition
  19. Product of primitive polynomials is primitive statement
  20. Gauss's lemma statement
  21. Eisenstein's criterion statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Eisenstein%27s_criterion, Criterion
  2. Web https://en.wikipedia.org/wiki/Eisenstein%27s_criterion, History
About this page
Tag02BPKD7TMD
URIm2://02BPKD7TMD
Kindstatement, theorem
WritingW1 stated
MSC12E05
StatusThis copy of the site has no build data. Run m2 status 02BPKD7TMD.
UsesIrreducible polynomial, Gauss's lemma, Primitive polynomial
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