Math 2.0

Commutative algebra Definition

Primitive polynomial

Builds on Content of a polynomial

Definition

Let RR be a commutative ring. A polynomial f∈R[x]f \in R[x] is primitive if every r∈Rr \in R that divides ff in R[x]R[x] is a unit. Here rr divides ff when rr divides each coefficient of ff.

Remarks

If RR is a unique factorisation domain, then ff is primitive if and only if its content is a unit. In ℤ[x]\mathbb{Z}[x], 3x2+2x−13x^2 + 2x - 1 is primitive and 6x2+4x−26x^2 + 4x - 2 is not. Every monic polynomial is primitive. The word primitive polynomial has another meaning over a finite field: a polynomial whose root is a primitive element.

Reading path

To understand this page from the ground up, read these 13 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…definitionPrimitive polynomial, definition, no evidence levelPrimitive polynomialdefinition
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. 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
  13. Primitive polynomial definition

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Primitive_part_and_content, lead section
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DefinesPrimitive polynomial
UsesContent of a polynomial

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