Math 2.0

Commutative algebra Definition

Content of a polynomial

Builds on Unique factorisation domainGreatest common divisor in an integral domain

Definition

Let RR be a unique factorisation domain and let f=a0+a1x+⋯+anxn∈R[x]f = a_0 + a_1 x + \dots + a_n x^n \in R[x]. The content of ff, written c(f)c(f), is a greatest common divisor of the coefficients a0,a1,…,ana_0, a_1, \dots, a_n.

Remarks

The content is unique up to associates. Over ℤ\mathbb{Z}, one often takes the positive gcd, with the sign of the leading coefficient in the primitive part. For f=6x2+4x−2f = 6x^2 + 4x - 2 in ℤ[x]\mathbb{Z}[x], c(f)=2c(f) = 2. Each nonzero ff is c(f)⋅gc(f) \cdot g with gg a primitive polynomial, the primitive part of ff. The content is multiplicative up to units: c(fg)∼c(f)c(g)c(f g) \sim c(f) c(g). This is a form of Gauss's lemma.

Reading path

To understand this page from the ground up, read these 12 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 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…definitionIntegral domain, definition, no evidence levelIntegral domaindefinitionAssociate elements, definition, no evidence levelAssociate elementsdefinitionIrreducible 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 …definitionContent of a polynomial, definition, no evidence levelContent of a polynomi…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.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Unit of a ring definition
  5. Zero divisor definition
  6. Divisibility in a commutative ring definition
  7. Integral domain definition
  8. Associate elements definition
  9. Irreducible element definition
  10. Greatest common divisor in an integral domain definition
  11. Unique factorisation domain definition
  12. Content of a polynomial definition

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Primitive_part_and_content, lead section
About this page
Tag08VV7YJRB7
URIm2://08VV7YJRB7
Kinddefinition
WritingW0 stub
MSC13F20
DefinesContent of a polynomial
UsesUnique factorisation domain, Greatest common divisor in an integral domain

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