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.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Witt ring | 2 |
| Witt ring of a field | 2 |
| Grothendieck–Witt group | 1 |
| Witt group | 1 |
| Witt group of quadratic forms canonical | 1 |
| Witt ring of Q | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8733410 — 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: Witt group of quadratic forms Context triple: [Hasse invariant, relatedTo, Witt group of quadratic forms]
-
A.
Rational Quadratic Forms
Rational Quadratic Forms is a classic monograph in number theory that systematically develops the arithmetic theory of quadratic forms over the rational numbers.
-
B.
Hermitian forms (work on quadratic forms)
Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
-
C.
Milnor K-theory
Milnor K-theory is an algebraic K-theory constructed from fields using tensor powers of their multiplicative groups modulo Steinberg relations, playing a central role in modern algebraic geometry and number theory.
-
D.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
-
E.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
- 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: Witt group of quadratic forms Target entity description: 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.
-
A.
Rational Quadratic Forms
Rational Quadratic Forms is a classic monograph in number theory that systematically develops the arithmetic theory of quadratic forms over the rational numbers.
-
B.
Hermitian forms (work on quadratic forms)
Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
-
C.
Milnor K-theory
Milnor K-theory is an algebraic K-theory constructed from fields using tensor powers of their multiplicative groups modulo Steinberg relations, playing a central role in modern algebraic geometry and number theory.
-
D.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
-
E.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
- F. None of above. chosen
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
ⓘ
linked to:
Witt group of quadratic forms
|
| 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 ⓘ |
| 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
ⓘ
Witt ring ⓘ
linked to:
Witt group of quadratic forms
algebraic K-theory ⓘ
linked to:
K-theory
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 ⓘ algebraic topology ⓘ classification of quadratic forms over fields ⓘ |
| usedToDefine |
Witt ring of a field
ⓘ
linked to:
Witt group of quadratic forms
|
How these facts were elicited
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Instruction
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Input
Subject: Witt group of quadratic forms Description of subject: 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.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Witt group of quadratic forms
linked to: Witt group of quadratic forms
linked to: Witt group of quadratic forms
linked to: Witt group of quadratic forms
linked to: Witt group of quadratic forms
linked to: Witt group of quadratic forms
linked to: Witt group of quadratic forms