Galois cohomology
E839567
Galois cohomology is a branch of mathematics that studies Galois groups and their actions on modules using cohomological methods, providing powerful tools for understanding field extensions, algebraic number theory, and arithmetic geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Galois cohomology canonical | 13 |
How this entity was disambiguated
This entity first appeared as the object of triple T10063343 — 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: Galois cohomology Context triple: [Hasse norm theorem, hasFormulationIn, Galois cohomology]
-
A.
Cohomologie Galoisienne
Cohomologie Galoisienne is a foundational monograph by Jean-Pierre Serre that systematically develops Galois cohomology and its deep applications in number theory and algebraic geometry.
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B.
Galois representations
Galois representations are homomorphisms from Galois groups of field extensions into matrix groups that encode deep arithmetic information and link number theory with algebraic geometry and modular forms.
-
C.
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.
-
D.
Galois theory
Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
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E.
étale cohomology
Étale cohomology is a cohomology theory in algebraic geometry that allows one to apply topological and cohomological methods to schemes, particularly over fields with nontrivial arithmetic such as finite fields.
- 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: Galois cohomology Target entity description: Galois cohomology is a branch of mathematics that studies Galois groups and their actions on modules using cohomological methods, providing powerful tools for understanding field extensions, algebraic number theory, and arithmetic geometry.
-
A.
Cohomologie Galoisienne
Cohomologie Galoisienne is a foundational monograph by Jean-Pierre Serre that systematically develops Galois cohomology and its deep applications in number theory and algebraic geometry.
-
B.
Galois representations
Galois representations are homomorphisms from Galois groups of field extensions into matrix groups that encode deep arithmetic information and link number theory with algebraic geometry and modular forms.
-
C.
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.
-
D.
Galois theory
Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
-
E.
étale cohomology
Étale cohomology is a cohomology theory in algebraic geometry that allows one to apply topological and cohomological methods to schemes, particularly over fields with nontrivial arithmetic such as finite fields.
- F. None of above. chosen
Statements (60)
| Predicate | Object |
|---|---|
| instanceOf |
branch of mathematics
ⓘ
cohomology theory ⓘ |
| appliesTo |
absolute Galois groups
ⓘ
algebraic extensions of fields ⓘ algebraic number fields ⓘ field extensions ⓘ global fields ⓘ local fields ⓘ |
| field |
algebra
ⓘ
arithmetic geometry ⓘ number theory ⓘ |
| formalDefinition | right derived functors of the fixed-point functor for Galois modules ⓘ |
| hasCanonicalReference |
Serre: Galois Cohomology
ⓘ
linked to:
Cohomologie Galoisienne
|
| hasHistoricalDevelopmentBy |
Chevalley
ⓘ
Serre ⓘ
linked to:
Jean-Pierre Serre
Tate ⓘ |
| hasKeyConcept |
Brauer group
ⓘ
Galois module ⓘ Hilbert 90 ⓘ Hochschild–Serre spectral sequence ⓘ Kummer theory ⓘ Poitou–Tate duality ⓘ Shapiro lemma ⓘ Tate cohomology ⓘ Tate–Shafarevich group ⓘ cohomological dimension of a field ⓘ cohomology group H^n(G,M) ⓘ continuous cochains ⓘ corestriction map ⓘ cup product ⓘ fundamental class in H^2 ⓘ inflation–restriction sequence ⓘ local Tate duality ⓘ
linked to:
Poitou–Tate duality
norm map ⓘ profinite Galois group ⓘ restriction map ⓘ |
| relatedTo |
Galois representations
ⓘ
K-theory of fields ⓘ
linked to:
K-theory
Selmer group ⓘ Weil group ⓘ class field theory ⓘ motivic cohomology ⓘ étale cohomology ⓘ |
| studies |
Galois groups
ⓘ
actions of Galois groups on modules ⓘ continuous group cohomology of Galois groups ⓘ |
| typicalGroup | absolute Galois group of a field ⓘ |
| typicalModule |
discrete Galois module
ⓘ
finite Galois module ⓘ p-adic Galois representation ⓘ |
| usedIn |
classification of central simple algebras
ⓘ
description of the Brauer group of a field ⓘ obstructions to local-global principles ⓘ study of principal homogeneous spaces ⓘ study of rational points on varieties ⓘ study of torsors under algebraic groups ⓘ |
| usesMethod |
Ext functors
ⓘ
derived functors ⓘ group cohomology ⓘ homological algebra ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Galois cohomology Description of subject: Galois cohomology is a branch of mathematics that studies Galois groups and their actions on modules using cohomological methods, providing powerful tools for understanding field extensions, algebraic number theory, and arithmetic geometry.
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.