Hurwitz theorem (composition algebras)
E931272
Hurwitz theorem (composition algebras) is a fundamental result in algebra that classifies all finite-dimensional normed division algebras over the real numbers, showing that they exist only in dimensions 1, 2, 4, and 8.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in normed division algebras ⓘ theorem in algebra ⓘ |
| allowsDimensions |
1
ⓘ
2 ⓘ 4 ⓘ 8 ⓘ |
| characterizes |
C as the unique 2-dimensional real normed division algebra up to isomorphism
ⓘ
H as the unique 4-dimensional real normed division algebra up to isomorphism ⓘ O as the unique 8-dimensional real normed division algebra up to isomorphism ⓘ R as the unique 1-dimensional real normed division algebra ⓘ |
| classifies |
finite-dimensional real normed division algebras
ⓘ
real composition algebras with multiplicative norm ⓘ |
| concerns |
composition algebra
ⓘ
multiplicative norm ⓘ normed division algebra ⓘ quadratic form ⓘ real numbers ⓘ |
| describes |
possible dimensions of composition algebras over the reals
ⓘ
structure of real normed division algebras ⓘ |
| equivalentTo | classification of real composition algebras with nondegenerate multiplicative quadratic form ⓘ |
| field |
algebra
ⓘ
number theory ⓘ topology ⓘ |
| forbidsDimensions | any other positive integer dimension than 1, 2, 4, or 8 for real normed division algebras ⓘ |
| hasConsequence |
existence of normed bilinear maps R^n × R^n → R^n only for n = 1, 2, 4, 8
ⓘ
restriction on possible dimensions of parallelizable spheres ⓘ |
| hasVariant | Hurwitz theorem on composition of quadratic forms NERFINISHED ⓘ |
| historicalPeriod | late 19th century ⓘ |
| implies |
every finite-dimensional real normed division algebra is isomorphic to R, C, H, or O as a normed algebra
ⓘ
there are no real normed division algebras of dimension 3 ⓘ there are no real normed division algebras of dimension 5 ⓘ there are no real normed division algebras of dimension 6 ⓘ there are no real normed division algebras of dimension 7 ⓘ there are no real normed division algebras of dimension greater than 8 ⓘ |
| impliesExistenceOf | only four isomorphism classes of real normed division algebras ⓘ |
| namedAfter | Adolf Hurwitz NERFINISHED ⓘ |
| provedBy | Adolf Hurwitz NERFINISHED ⓘ |
| relatedTo |
complex numbers C
ⓘ
octonions O ⓘ quaternions H ⓘ real numbers R ⓘ |
| statement |
every finite-dimensional real normed division algebra has dimension 1, 2, 4, or 8
ⓘ
the only finite-dimensional real composition algebras with a positive-definite multiplicative norm have dimensions 1, 2, 4, or 8 ⓘ |
| usesConcept |
bilinear form
ⓘ
composition of quadratic forms ⓘ quadratic form identity ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Adolf Hurwitz