Math 2.0

statement theorem

Polynomial ring over a UFD is a UFD

Statement

Let RR be a unique factorisation domain. Then the polynomial ring R[x]R[x] is a unique factorisation domain.

Remarks

The proof uses Gauss's lemma and unique factorisation in K[x]K[x], where KK is the field of fractions of RR. The irreducible elements of R[x]R[x] are the irreducible elements of RR and the primitive polynomials of positive degree that are irreducible in K[x]K[x]. By induction, R[x1,…,xn]R[x_1, \dots, x_n] is a unique factorisation domain. In particular, Z[x]\mathbb{Z}[x] and K[x,y]K[x, y] are. These rings are not principal ideal domains: (2,x)(2, x) in Z[x]\mathbb{Z}[x] and (x,y)(x, y) in K[x,y]K[x, y] are not principal.

Explanations

L1 human

Integer polynomials factor uniquely into irreducible pieces, like integers do. The pieces are prime numbers and irreducible primitive polynomials. For example, 2x2−2=2(x−1)(x+1)2x^2 - 2 = 2 (x - 1)(x + 1).

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To read this node, read these 21 nodes in this order. The foundations come first, and the list ends with this node.

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 lemmastatementPolynomial ring over a UFD is a UFD, statement, no evidence levelPolynomial ring over …statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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. 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. Polynomial ring over a UFD is a UFD statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Unique_factorization_domain, Examples

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