Hurwitz quaternions
E620663
Dedekind domain
Euclidean domain
lattice
maximal order
noncommutative ring
principal ideal domain
quaternion order
Hurwitz quaternions are a specific lattice of quaternions with integer and half-integer components that form a maximal order in the quaternion algebra and provide a natural algebraic framework for understanding representations of integers as sums of four squares.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Dedekind domain
ⓘ
Euclidean domain ⓘ lattice ⓘ maximal order ⓘ noncommutative ring ⓘ principal ideal domain ⓘ quaternion order ⓘ |
| closedUnder |
addition
ⓘ
conjugation ⓘ multiplication ⓘ |
| contains | Lipschitz quaternions NERFINISHED ⓘ |
| definedOver | rational numbers ⓘ |
| formsLatticeIn | R^4 ⓘ |
| hasAlternativeName |
Hurwitz integral quaternions
NERFINISHED
ⓘ
Hurwitz order NERFINISHED ⓘ |
| hasBasis |
{(1+i+j+k)/2, i, j, k}
ⓘ
{1, i, j, k} ⓘ |
| hasCenter | integers ⓘ |
| hasComponentType |
half-integers
ⓘ
integers ⓘ |
| hasDiscriminant | 2 ⓘ |
| hasNormForm | sum of four squares ⓘ |
| hasRankAsZModule | 4 ⓘ |
| hasUniqueFactorizationOfElementsUpToUnits | true ⓘ |
| hasUniqueFactorizationOfIdeals | true ⓘ |
| hasUnitGroup | binary tetrahedral group NERFINISHED ⓘ |
| hasZeroDivisors | false ⓘ |
| introducedBy | Adolf Hurwitz NERFINISHED ⓘ |
| isFiniteOver | integers ⓘ |
| isFreeZModuleOfRank | 4 ⓘ |
| isIntegralDomain | false ⓘ |
| isLeftEuclidean | true ⓘ |
| isMaximalOrderIn | Hamilton quaternions over Q NERFINISHED ⓘ |
| isNoncommutative | true ⓘ |
| isOrderIn | Hamilton quaternion algebra over Q ⓘ |
| isRightEuclidean | true ⓘ |
| isSubsetOf | Hamilton quaternions ⓘ |
| isSupersetOf | Lipschitz quaternions NERFINISHED ⓘ |
| normIsMultiplicative | true ⓘ |
| normMap | maps to nonnegative integers ⓘ |
| providesFrameworkFor | representations of integers as sums of four squares ⓘ |
| relatedTo |
ADE classification
ⓘ
D4 root lattice ⓘ |
| unitGroupOrder | 24 ⓘ |
| usedIn |
arithmetic of quaternion algebras
ⓘ
sphere packings in four dimensions ⓘ |
| usedToProve | Lagrange four-square theorem ⓘ |
| yearIntroducedApprox | 1919 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.