Witt group of quadratic forms
E753152
The Witt group of quadratic forms is an algebraic structure that classifies nondegenerate quadratic forms over a field up to stable equivalence, with addition given by orthogonal sum and inverses given by taking opposite forms.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Witt group | 1 |
| Witt ring | 1 |
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
Witt group
ⓘ
algebraic structure ⓘ group ⓘ invariant of quadratic forms ⓘ |
| arisesIn |
study of quadratic forms over global fields
ⓘ
study of quadratic forms over local fields ⓘ |
| captures | anisotropic part of quadratic forms ⓘ |
| classifies | nondegenerate quadratic forms over a field up to stable equivalence ⓘ |
| constructedBy | Grothendieck group completion of the monoid of quadratic forms modulo hyperbolic forms ⓘ |
| constructedFrom | monoid of isometry classes of nondegenerate quadratic forms ⓘ |
| definedOver | field ⓘ |
| dependsOn | base field ⓘ |
| elementType | equivalence class of nondegenerate quadratic forms ⓘ |
| encodes | isometry classes of quadratic forms modulo hyperbolic summands ⓘ |
| equivalenceRelation | stable equivalence via addition of hyperbolic forms ⓘ |
| generalizes | classification of quadratic forms over the real numbers by signature ⓘ |
| hasFunctoriality |
contravariant in field homomorphisms
ⓘ
covariant for field extensions via scalar extension ⓘ |
| hasGroupOperation | orthogonal sum ⓘ |
| hasHomomorphismTo | Grothendieck–Witt group NERFINISHED ⓘ |
| hasIdentityElement |
class of hyperbolic quadratic forms
ⓘ
class of zero form in the Witt group ⓘ |
| hasInverseOperation | taking opposite quadratic form ⓘ |
| hasOperation | orthogonal sum of quadratic forms ⓘ |
| hasProperty | commutative group under orthogonal sum ⓘ |
| hasTrivialGroup | for algebraically closed fields of characteristic not 2 ⓘ |
| introducedBy | Ernst Witt NERFINISHED ⓘ |
| kernelOf | rank and discriminant invariants in some cases ⓘ |
| notation | W(F) ⓘ |
| parameterizedBy | field F ⓘ |
| quotientsOut |
hyperbolic quadratic forms
ⓘ
metabolic quadratic forms ⓘ |
| relatedConcept |
Witt decomposition of quadratic forms
ⓘ
Witt index of a quadratic form ⓘ |
| relatedTo |
Milnor K-theory
NERFINISHED
ⓘ
Witt ring NERFINISHED ⓘ algebraic K-theory NERFINISHED ⓘ signature homomorphism for real closed fields ⓘ symmetric bilinear forms ⓘ |
| requires | nondegeneracy of quadratic forms ⓘ |
| sensitiveTo | characteristic of the field ⓘ |
| usedIn |
algebraic geometry
ⓘ
algebraic number theory NERFINISHED ⓘ algebraic topology ⓘ classification of quadratic forms over fields ⓘ |
| usedToDefine | Witt ring of a field NERFINISHED ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Witt ring
this entity surface form:
Witt group