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.
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
→
|
| 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 → |
Referenced by (2)
| Subject (surface form when different) | Predicate |
|---|---|
|
Carl Friedrich Gauss
→
|
hasConceptNamedAfter |
|
Ulam spiral
→
|
relatedConcept |