Math 2.0

Commutative algebra Theorem

Bézout's identity in a Euclidean domain

Builds on Euclidean domainGreatest common divisor in an integral domainEuclidean algorithm in a Euclidean domain

Statement

Let RR be a Euclidean domain and let a,b∈Ra, b \in R. Then aa and bb have a greatest common divisor dd, and there are s,t∈Rs, t \in R with d=sa+tb.d = s a + t b.

Remarks

The Euclidean algorithm computes dd. The extended Euclidean algorithm also computes ss and tt, by back substitution. In ℤ\mathbb{Z}: gcd⁡(252,198)=18=4⋅252−5⋅198\gcd(252, 198) = 18 = 4 \cdot 252 - 5 \cdot 198. The same identity holds in every principal ideal domain, because the ideal (a,b)(a, b) is principal. If d≠0d \neq 0, the coefficients are not unique: s+kb/ds + k b/d and t−ka/dt - k a/d also work for each k∈Rk \in R.

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 elementsdefinitionEuclidean domain, definition, no evidence levelEuclidean domaindefinitionGreatest common divisor in an integral domain, definition, no evidence levelGreatest common divis…definitionEuclidean algorithm in a Euclidean domain, concept, no evidence levelEuclidean algorithm i…conceptBézout's identity in a Euclidean domain, statement, no evidence levelBézout's identity in …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. 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. Euclidean domain definition
  10. Greatest common divisor in an integral domain definition
  11. Euclidean algorithm in a Euclidean domain concept
  12. Bézout's identity in a Euclidean domain statement

Further reading

  1. Web https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity, For principal ideal domains
About this page
Tag0VY3WPP9Y3
URIm2://0VY3WPP9Y3
Kindstatement, theorem
WritingW0 stub
MSC13F07
StatusThis copy of the site has no build data. Run m2 status 0VY3WPP9Y3.
UsesEuclidean domain, Greatest common divisor in an integral domain, Euclidean algorithm in a Euclidean domain

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