Euclidean domains

E627994

algebraic structure commutative ring integral domain

Euclidean domains are a class of integral domains that admit a division algorithm based on a Euclidean function, generalizing the arithmetic of the integers and enabling efficient computation of greatest common divisors.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Euclidean domain	0

Statements (50)

Predicate	Object
instanceOf	algebraic structure ⓘ commutative ring ⓘ integral domain ⓘ
definedOn	commutative ring with identity ⓘ
generalizes	arithmetic of the integers ⓘ division algorithm in ℤ ⓘ
hasAlternativeName	Euclidean ring ⓘ
hasConstraint	Euclidean function is not unique ⓘ not every principal ideal domain is Euclidean ⓘ
hasDefinitionComponent	Euclidean function ⓘ division with remainder ⓘ well‑founded measure on nonzero elements ⓘ
hasExample	Eisenstein integers ℤ[ω] with ω = e^{2πi/3} ⓘ Gaussian integers ℤ[i] NERFINISHED ⓘ localization of a Euclidean domain at a multiplicative set ⓘ polynomial ring F[x] over a field F ⓘ ring of integers ℤ ⓘ ring of integers ℤ[√2] ⓘ ring of polynomials K[x] over any Euclidean domain K ⓘ
hasHistoricalOrigin	generalization of Euclid’s algorithm for integers ⓘ
hasNonExample	ring of integers of ℚ(√−5) ⓘ ring of integers ℤ[(1+√−19)/2] ⓘ
hasOpenProblem	classification of number fields with Euclidean ring of integers ⓘ
hasProperty	Noetherian ⓘ admits division algorithm ⓘ every ideal is principal ⓘ integrally closed in its field of fractions ⓘ supports Euclidean algorithm for gcd ⓘ
implies	Bezout identity for gcds ⓘ every irreducible element is prime ⓘ every nonzero ideal is generated by a single element ⓘ existence of gcd for any pair of elements ⓘ gcd can be computed by Euclidean algorithm ⓘ unique factorization into irreducibles ⓘ
relatedTo	Bezout domain NERFINISHED ⓘ Dedekind domain ⓘ norm-Euclidean domain ⓘ
requires	existence of Euclidean function d: R\{0} → ℕ ⓘ for all a,b in R with b≠0, there exist q,r in R such that a = bq + r and (r = 0 or d(r) < d(b)) ⓘ for all a,b≠0 in R, d(ab) ≥ d(a) ⓘ integral domain property ⓘ
subclassOf	principal ideal domain ⓘ unique factorization domain ⓘ
usedFor	computing Bezout coefficients ⓘ efficient computation of greatest common divisors ⓘ extended Euclidean algorithm NERFINISHED ⓘ ideal-theoretic computations ⓘ
usedIn	algebraic number theory NERFINISHED ⓘ commutative algebra ⓘ computational algebra ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

ring theory → usesConcept → Euclidean domains ⓘ