Math 2.0

Commutative algebra Definition

Factorization of polynomials

Builds on FieldPolynomial ringIrreducible polynomialMonic polynomialUnique factorisation domain

Definition

Let KK be a field and let f∈K[x]f \in K[x] be a nonzero polynomial. A factorization of ff into irreducibles is an expression f=c p1e1⋯pkekf = c \, p_1^{e_1} \cdots p_k^{e_k} such that:

  1. c∈Kc \in K is nonzero;
  2. p1,…,pkp_1, \dots, p_k are pairwise distinct monic irreducible polynomials in K[x]K[x], with k≥0k \ge 0;
  3. e1,…,eke_1, \dots, e_k are positive integers.

Remarks

Every nonzero ff has such a factorization, and it is unique up to the order of the factors, since K[x]K[x] is a unique factorisation domain. The constant cc is the leading coefficient of ff. Example: over ℚ\Q, 2x3−2x=2 x(x−1)(x+1)2x^3 - 2x = 2 \, x (x - 1)(x + 1); each factor is monic of degree 11, hence irreducible. Non-example: x4−1=(x2−1)(x2+1)x^4 - 1 = (x^2 - 1)(x^2 + 1) is a factorization but not into irreducibles, since x2−1=(x−1)(x+1)x^2 - 1 = (x - 1)(x + 1) violates clause 2. Over a unique factorisation domain RR such as ℤ\Z, a nonzero f∈R[x]f \in R[x] is its content times a primitive polynomial. The primitive part factors into primitive irreducibles, by polynomial ring over a UFD is a UFD. Over finite fields, see factorization over finite fields.

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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 coefficientdefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionUnique factorisation domain, definition, no evidence levelUnique factorisation …definitionFactorization of polynomials, definition, no evidence levelFactorization of poly…definition
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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. Degree of a polynomial definition
  9. Field definition
  10. Integral domain definition
  11. Associate elements definition
  12. Leading coefficient definition
  13. Irreducible element definition
  14. Monic polynomial definition
  15. Irreducible polynomial definition
  16. Unique factorisation domain definition
  17. Factorization of polynomials definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Factorization_of_polynomials
About this page
TagEFDBS6ZME8
URIm2://EFDBS6ZME8
Kinddefinition
WritingW0 stub
MSC13P05, 12E05, 12D05
DefinesFactorization of polynomials
UsesField, Polynomial ring, Irreducible polynomial, Monic polynomial, Unique factorisation domain
Changed2 November 2026

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