Herbrand function
E753159
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Herbrand function canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8733558 — 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 function Context triple: [Hasse–Arf theorem, relatedTo, Herbrand function]
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A.
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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B.
Herbrand interpretation
A Herbrand interpretation is a foundational model-theoretic construct in logic and automated theorem proving that interprets formulas over the Herbrand universe built from a theory’s own function symbols and constants.
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C.
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.
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D.
Herbrand expansion
Herbrand expansion is a method in mathematical logic that transforms first-order formulas into equivalent (often infinite) propositional combinations by systematically instantiating quantified variables with terms from the Herbrand universe.
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E.
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.
- 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 function Target entity description: 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.
-
A.
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.
-
B.
Herbrand interpretation
A Herbrand interpretation is a foundational model-theoretic construct in logic and automated theorem proving that interprets formulas over the Herbrand universe built from a theory’s own function symbols and constants.
-
C.
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.
-
D.
Herbrand expansion
Herbrand expansion is a method in mathematical logic that transforms first-order formulas into equivalent (often infinite) propositional combinations by systematically instantiating quantified variables with terms from the Herbrand universe.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
function in number theory
ⓘ
mathematical concept ⓘ tool in local class field theory ⓘ |
| appearsIn |
Serre Local Fields
ⓘ
classical local class field theory ⓘ |
| appliesTo | Galois group of a finite Galois extension of local fields ⓘ |
| associatedWith |
ramification groups in the lower numbering
ⓘ
ramification groups in the upper numbering ⓘ |
| characterizes | ramification filtration of Galois groups ⓘ |
| codomain | real numbers ⓘ |
| context |
finite Galois extensions of non-archimedean local fields
ⓘ
ramified extensions of p-adic fields ⓘ |
| dependsOn | sizes of ramification subgroups ⓘ |
| domain |
Galois extensions of local fields
ⓘ
non-archimedean local fields ⓘ |
| ensures | functorial behavior of upper ramification filtration under quotients ⓘ |
| field |
local class field theory
ⓘ
number theory ⓘ |
| formalVariable | real parameter u ≥ -1 ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| input | real parameter describing lower-numbered ramification groups ⓘ |
| measures | ramification behavior in extensions of local fields ⓘ |
| namedAfter | Jacques Herbrand ⓘ |
| output | real parameter describing upper-numbered ramification groups ⓘ |
| property |
breakpoints determined by jumps in ramification filtration
ⓘ
continuous from the right ⓘ increasing function of a real variable ⓘ piecewise linear ⓘ |
| relatedConcept |
Herbrand quotient
ⓘ
lower numbering of ramification groups ⓘ ramification filtration ⓘ upper numbering of ramification groups ⓘ |
| relatedTo |
different of an extension of local fields
ⓘ
discriminant of an extension of local fields ⓘ |
| relates |
lower numbering of ramification groups
ⓘ
upper numbering of ramification groups ⓘ |
| role |
normalizes ramification filtration to behave well under quotients
ⓘ
transfers the filtration index from lower to upper numbering ⓘ |
| usedBy |
number theorists studying local Galois representations
ⓘ
researchers in arithmetic geometry ⓘ |
| usedFor |
comparing ramification in different extensions
ⓘ
defining upper ramification breaks ⓘ describing wild ramification ⓘ reindexing ramification groups from lower to upper numbering ⓘ |
| usedIn |
Galois theory of local fields
ⓘ
local class field theory ⓘ ramification theory ⓘ |
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 function Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.