Math 2.0

Number theory Theorem

Irreducibility of the cyclotomic polynomials over the rationals

Builds on Cyclotomic polynomialIrreducible polynomial

Statement

For every integer n≥1n \ge 1, the cyclotomic polynomial Φn\Phi_n is irreducible in ℚ[x]\mathbb{Q}[x].

Remarks

For n=pn = p prime, Eisenstein's criterion applied to Φp(x+1)\Phi_p(x + 1) with the prime pp proves the statement. The general case is harder. One proof shows that if ζ\zeta is a root of an irreducible factor of Φn\Phi_n, then so is ζp\zeta^p for each prime p∤np \nmid n. A consequence: Φn\Phi_n is the minimal polynomial over ℚ\mathbb{Q} of each primitive nn-th root of unity. So the degree of ℚ(e2πi/n)\mathbb{Q}(e^{2\pi i/n}) over ℚ\mathbb{Q} is φ(n)\varphi(n).

Reading path

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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 divisordefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionCyclotomic polynomial, definition, no evidence levelCyclotomic polynomialdefinitionIrreducibility of the cyclotomic polynomials over the rationals, statement, no evidence levelIrreducibility of the…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.
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  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. Degree of a polynomial definition
  8. Integral domain definition
  9. Leading coefficient definition
  10. Irreducible element definition
  11. Monic polynomial definition
  12. Irreducible polynomial definition
  13. Cyclotomic polynomial definition
  14. Irreducibility of the cyclotomic polynomials over the rationals statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclotomic_polynomial, Properties
About this page
Tag0SK1VHZ5R2
URIm2://0SK1VHZ5R2
Kindstatement, theorem
WritingW0 stub
MSC11R18
StatusThis copy of the site has no build data. Run m2 status 0SK1VHZ5R2.
UsesCyclotomic polynomial, Irreducible polynomial

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