Gaussian integers
E29369
Euclidean domain
integral domain
lattice in the complex plane
mathematical object
principal ideal domain
ring
unique factorization domain
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gaussian integers canonical | 2 |
| Gaussian integers ℤ[i] | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T228959 — 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: Gaussian integers Context triple: [Carl Friedrich Gauss, hasConceptNamedAfter, Gaussian integers]
-
A.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
B.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
C.
Numbers
Numbers is the fourth book of the Hebrew Bible and the Christian Old Testament, recounting the Israelites’ wilderness wanderings and organizing laws and censuses.
-
D.
Minkowski sum
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
-
E.
Leopold Kronecker
Leopold Kronecker was a 19th-century German mathematician known for his work in number theory, algebra, and logic, and for his influential finitist and constructivist views on mathematics.
- 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: Gaussian integers Target entity description: 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.
-
A.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
B.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
-
C.
Numbers
Numbers is the fourth book of the Hebrew Bible and the Christian Old Testament, recounting the Israelites’ wilderness wanderings and organizing laws and censuses.
-
D.
Minkowski sum
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
-
E.
Leopold Kronecker
Leopold Kronecker was a 19th-century German mathematician known for his work in number theory, algebra, and logic, and for his influential finitist and constructivist views on mathematics.
- F. None of above. chosen
Statements (60)
| Predicate | Object |
|---|---|
| instanceOf |
Euclidean domain
ⓘ
integral domain ⓘ lattice in the complex plane ⓘ mathematical object ⓘ principal ideal domain ⓘ ring ⓘ unique factorization domain ⓘ |
| associatedWith |
Fermat's sum of two squares theorem
ⓘ
linked to:
Fermat's theorem on sums of two squares
|
| characterizes | odd prime p is sum of two squares iff p is not prime in ℤ[i] ⓘ |
| closedUnder |
addition
ⓘ
multiplication ⓘ subtraction ⓘ |
| contains |
all numbers of the form a+0i with a in ℤ
ⓘ
ordinary integers ⓘ |
| definedAs | complex numbers of the form a+bi where a and b are integers ⓘ |
| distanceMetric | induced by complex absolute value ⓘ |
| EuclideanFunction | norm ⓘ |
| fieldOfFractions | Gaussian rationals ℚ(i) ⓘ |
| forms | two-dimensional lattice over ℤ ⓘ |
| geometricStructure | square lattice in the complex plane ⓘ |
| hasAdditiveGroupIsomorphicTo | ℤ² ⓘ |
| hasClassNumber | 1 ⓘ |
| hasElementForm | a+bi with a,b in ℤ and i² = -1 ⓘ |
| hasKrullDimension | 1 ⓘ |
| hasNormFunction | N(a+bi) = a² + b² ⓘ |
| hasPrimeElement |
1+i
ⓘ
1-i ⓘ 2 ⓘ 2+i ⓘ 2-i ⓘ 3+2i ⓘ 3-2i ⓘ 5 ⓘ |
| hasUnit |
-1
ⓘ
-i ⓘ 1 ⓘ i ⓘ |
| hasZeroDivisors | false ⓘ |
| isCommutativeRingWithIdentity | true ⓘ |
| isDedekindDomain | true ⓘ |
| isIntegrallyClosed | true ⓘ |
| isNoetherianRing | true ⓘ |
| isRingOfIntegersOf | quadratic field ℚ(i) ⓘ |
| maximalIdealsCorrespondTo | Gaussian primes ⓘ |
| normIs | multiplicative ⓘ |
| normTakesValuesIn | nonnegative integers ⓘ |
| notClosedUnder | division ⓘ |
| numberOfUnits | 4 ⓘ |
| primeFactorization | every nonzero nonunit factors uniquely up to units and order ⓘ |
| primeInGaussianIntegersCondition | odd prime p ≡ 3 (mod 4) remains prime in ℤ[i] ⓘ |
| primeSplittingProperty | odd prime p ≡ 1 (mod 4) factors as π·π̄ in ℤ[i] ⓘ |
| ramifiedPrime | 2 = (1+i)² up to units ⓘ |
| subsetOf |
complex numbers
ⓘ
ℂ ⓘ |
| symbol | ℤ[i] ⓘ |
| unitsFormGroupIsomorphicTo | cyclic group of order 4 ⓘ |
| usedIn |
algebraic geometry over ℤ[i]
ⓘ
algebraic number theory ⓘ lattice-based constructions in geometry of numbers ⓘ proofs of representation of primes as sums of two squares ⓘ |
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: Gaussian integers Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.