Herbrand quotient
E839565
The Herbrand quotient is an invariant in algebraic number theory and group cohomology that measures the relative sizes of certain cohomology groups associated with a finite group action on a module.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Herbrand quotient canonical | 3 |
| Herbrand cohomology | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10063339 — 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: Herbrand quotient Context triple: [Hasse norm theorem, relatedTo, Herbrand quotient]
-
A.
Herbrand's theorem
Herbrand's theorem is a fundamental result in mathematical logic and proof theory that characterizes the validity of first-order formulas via finite sets of ground instances, forming a basis for automated theorem proving.
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B.
Herbrand function
The Herbrand function is a numerical tool in local class field theory that measures the ramification filtration of Galois groups, playing a key role in understanding how ramification behaves in extensions of local fields.
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C.
Herbrand universe
The Herbrand universe is a fundamental concept in mathematical logic and automated theorem proving, consisting of all ground (variable-free) terms that can be built from the function symbols and constants of a given first-order language.
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D.
Herbrand semantics
Herbrand semantics is a formal framework in logic and automated theorem proving that interprets first-order formulas over the Herbrand universe of ground terms to define truth and satisfiability.
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E.
Herbrand disjunction
Herbrand disjunction is a logical formula formed as a finite disjunction of ground instances of a first-order formula, central to Herbrand’s theorem in proof theory and automated reasoning.
- 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: Herbrand quotient Target entity description: The Herbrand quotient is an invariant in algebraic number theory and group cohomology that measures the relative sizes of certain cohomology groups associated with a finite group action on a module.
-
A.
Herbrand's theorem
Herbrand's theorem is a fundamental result in mathematical logic and proof theory that characterizes the validity of first-order formulas via finite sets of ground instances, forming a basis for automated theorem proving.
-
B.
Herbrand function
The Herbrand function is a numerical tool in local class field theory that measures the ramification filtration of Galois groups, playing a key role in understanding how ramification behaves in extensions of local fields.
-
C.
Herbrand universe
The Herbrand universe is a fundamental concept in mathematical logic and automated theorem proving, consisting of all ground (variable-free) terms that can be built from the function symbols and constants of a given first-order language.
-
D.
Herbrand semantics
Herbrand semantics is a formal framework in logic and automated theorem proving that interprets first-order formulas over the Herbrand universe of ground terms to define truth and satisfiability.
-
E.
Herbrand disjunction
Herbrand disjunction is a logical formula formed as a finite disjunction of ground instances of a first-order formula, central to Herbrand’s theorem in proof theory and automated reasoning.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic number theory concept
ⓘ
cohomological invariant ⓘ group cohomology concept ⓘ |
| appearsIn |
cohomological formulas for class groups
ⓘ
cohomological formulas for unit groups ⓘ |
| appliesTo |
G-modules
ⓘ
finite group actions ⓘ |
| associatedWith |
H^0(G,M)
ⓘ
H^1(G,M) ⓘ |
| assumes |
G finite
ⓘ
M a finitely generated G-module in arithmetic applications ⓘ |
| category | mathematical invariant ⓘ |
| context | finite Galois extensions of number fields ⓘ |
| definedBy | Jacques Herbrand in the context of class field theory ⓘ |
| definedFor |
G-module M
ⓘ
finite group G ⓘ |
| domain | cohomology of groups ⓘ |
| field |
algebraic number theory
ⓘ
group cohomology ⓘ |
| formalism | group cohomology ⓘ |
| generalizes | index computations in cohomology ⓘ |
| hasFormula | h(G,M) = |H^0(G,M)| / |H^1(G,M)| when both groups are finite ⓘ |
| hasProperty |
equals 1 for many natural G-modules in number theory
ⓘ
multiplicative in short exact sequences of G-modules under finiteness conditions ⓘ |
| invariantOf | pair (G,M) ⓘ |
| invariantUnder | isomorphisms of G-modules ⓘ |
| measures | relative sizes of cohomology groups ⓘ |
| namedAfter | Jacques Herbrand ⓘ |
| relatedTo |
Euler characteristic in group cohomology
ⓘ
Herbrand’s theorem ⓘ
linked to:
Herbrand's theorem
Tate cohomology groups ⓘ |
| requires |
H^0(G,M) finite
ⓘ
H^1(G,M) finite ⓘ |
| toolFor | relating arithmetic invariants to cohomology groups ⓘ |
| usedIn |
Galois cohomology
ⓘ
Galois module structure of units ⓘ Tate cohomology theory ⓘ
linked to:
Tate cohomology
class field theory ⓘ
linked to:
global class field theory
proofs of relations between unit ranks and class numbers ⓘ study of ideal class groups ⓘ |
| usedToCompare | fixed points and coinvariants of G on M ⓘ |
| usedToStudy |
Galois action on ideal class groups
ⓘ
Galois action on units of number fields ⓘ |
| valuesIn | positive rational numbers ⓘ |
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: Herbrand quotient Description of subject: The Herbrand quotient is an invariant in algebraic number theory and group cohomology that measures the relative sizes of certain cohomology groups associated with a finite group action on a module.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Herbrand quotient