Euclidean domains
E627994
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Euclidean domains canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6908960 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Euclidean domains Context triple: [ring theory, usesConcept, Euclidean domains]
-
A.
Dedekind domain
A Dedekind domain is an integral domain in which every nonzero proper ideal factors uniquely into a product of prime ideals, playing a central role in algebraic number theory and the study of rings of integers in number fields.
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B.
Noetherian rings
Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
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C.
Gaussian integers
Gaussian integers are complex numbers whose real and imaginary parts are both integers, forming a lattice in the complex plane with important applications in number theory and algebra.
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D.
Gaussian rationals ℚ(i)
Gaussian rationals ℚ(i) are the field of complex numbers whose real and imaginary parts are rational, formed by adjoining the imaginary unit i to the rational numbers.
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E.
Henselian ring
A Henselian ring is a local ring in which Hensel’s lemma holds, allowing certain types of polynomial factorizations and root liftings from the residue field to the ring itself.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Euclidean domains Target entity description: 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.
-
A.
Dedekind domain
A Dedekind domain is an integral domain in which every nonzero proper ideal factors uniquely into a product of prime ideals, playing a central role in algebraic number theory and the study of rings of integers in number fields.
-
B.
Noetherian rings
Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
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C.
Gaussian integers
Gaussian integers are complex numbers whose real and imaginary parts are both integers, forming a lattice in the complex plane with important applications in number theory and algebra.
-
D.
Gaussian rationals ℚ(i)
Gaussian rationals ℚ(i) are the field of complex numbers whose real and imaginary parts are rational, formed by adjoining the imaginary unit i to the rational numbers.
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E.
Henselian ring
A Henselian ring is a local ring in which Hensel’s lemma holds, allowing certain types of polynomial factorizations and root liftings from the residue field to the ring itself.
- F. None of above. chosen
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] ⓘ
linked to:
Gaussian integers
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
ⓘ
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 ⓘ
linked to:
Euclidean algorithm for polynomials
ideal-theoretic computations ⓘ |
| usedIn |
algebraic number theory
ⓘ
commutative algebra ⓘ computational algebra ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Euclidean domains Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.