Math 2.0

Commutative algebra Definition

Polynomial greatest common divisor

Builds on FieldPolynomial ringMonic polynomialGreatest common divisor in an integral domain

Definition

Let KK be a field and let f,g∈K[x]f, g \in K[x] be polynomials that are not both 00. A greatest common divisor of ff and gg is a polynomial d∈K[x]d \in K[x] such that:

  1. dd divides ff and dd divides gg;
  2. every polynomial that divides both ff and gg divides dd.

The greatest common divisor gcd⁡(f,g)\gcd(f, g) is the unique monic polynomial with these two properties.

Remarks

Two greatest common divisors differ by a nonzero constant factor, so exactly one of them is monic. Example: gcd⁡(x2−1,x2−3x+2)=x−1\gcd(x^2 - 1, x^2 - 3x + 2) = x - 1. Check: x−1x - 1 divides both, and a common divisor divides (x2−1)−(x2−3x+2)=3x−3(x^2 - 1) - (x^2 - 3x + 2) = 3x - 3, hence divides x−1x - 1. Non-example: 2x−22x - 2 satisfies both clauses, but it is not monic, so it is a greatest common divisor and not gcd⁡(f,g)\gcd(f, g). The Euclidean algorithm computes gcd⁡(f,g)\gcd(f, g), and Bézout's identity gives a,b∈K[x]a, b \in K[x] with af+bg=gcd⁡(f,g)a f + b g = \gcd(f, g). Over a unique factorisation domain such as ℤ\Z, greatest common divisors in ℤ[x]\Z[x] also exist, but only up to units, see greatest common divisor in an integral domain.

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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 coefficientdefinitionGreatest common divisor in an integral domain, definition, no evidence levelGreatest common divis…definitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionPolynomial greatest common divisor, definition, no evidence levelPolynomial greatest c…definition
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. 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. Greatest common divisor in an integral domain definition
  14. Monic polynomial definition
  15. Polynomial greatest common divisor definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor
About this page
Tag99J1WXG3F7
URIm2://99J1WXG3F7
Kinddefinition
WritingW0 stub
MSC13F20, 13A05
DefinesPolynomial greatest common divisor
UsesField, Polynomial ring, Monic polynomial, Greatest common divisor in an integral domain
Changed2 November 2026

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