Math 2.0

Commutative algebra Theorem

K[x] is a Euclidean domain

Builds on Euclidean domainDivision algorithm for polynomialsDegree of a polynomialR[x] is an integral domain iff R is

Statement

Let KK be a field. Then the polynomial ring K[x]K[x] is a Euclidean domain, with the degree as Euclidean function. That is, for all f,g∈K[x]f, g \in K[x] with g≠0g \neq 0, there are q,r∈K[x]q, r \in K[x] with f=qg+rf = q g + r and r=0r = 0 or deg⁡r<deg⁡g\deg r < \deg g.

Remarks

The ring K[x]K[x] is an integral domain because KK is one. The existence of qq and rr is the division algorithm over a field. A consequence is that K[x]K[x] is a principal ideal domain and a unique factorization domain. The ring ℤ[x]\mathbb{Z}[x] is not a Euclidean domain, because it is not a principal ideal domain.

Reading path

To understand this page from the ground up, read these 14 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 ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionEuclidean domain, definition, no evidence levelEuclidean domaindefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionDegree of a product, statement, no evidence levelDegree of a productstatementDivision algorithm for polynomials, statement, no evidence levelDivision algorithm fo…statementR[x] is an integral domain iff R is, statement, no evidence levelR[x] is an integral d…statementK[x] is a Euclidean domain, statement, no evidence levelK[x] is a Euclidean d…statement
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. Polynomial ring definition
  5. Zero divisor definition
  6. Degree of a polynomial definition
  7. Integral domain definition
  8. Leading coefficient definition
  9. Euclidean domain definition
  10. Monic polynomial definition
  11. Degree of a product statement
  12. Division algorithm for polynomials statement
  13. R[x] is an integral domain iff R is statement
  14. K[x] is a Euclidean domain statement

Where this leads

22 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Euclidean_domain, Examples
About this page
Tag0DQG39K2NH
URIm2://0DQG39K2NH
Kindstatement, theorem
WritingW0 stub
MSC13F07
StatusThis copy of the site has no build data. Run m2 status 0DQG39K2NH.
UsesEuclidean domain, Division algorithm for polynomials, Degree of a polynomial, R[x] is an integral domain iff R is

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