Hasse–Arf theorem
E207315
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hasse–Arf theorem canonical | 4 |
| Hasse–Arf theorem (in the abelian case) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1862418 — 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: Hasse–Arf theorem Context triple: [Helmut Hasse, notableWork, Hasse–Arf theorem]
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A.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
B.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
C.
Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
-
D.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
E.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
- 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: Hasse–Arf theorem Target entity description: The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
-
A.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
B.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
C.
Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
-
D.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
E.
Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in local class field theory ⓘ theorem in algebraic number theory ⓘ |
| appliesTo |
Galois groups of abelian extensions of local fields
ⓘ
finite abelian extensions of local fields ⓘ |
| asserts | jumps in the upper numbering ramification filtration of abelian extensions occur at integers ⓘ |
| clarifies | structure of higher ramification groups in abelian extensions ⓘ |
| concerns |
abelian extensions of local fields
ⓘ
jumps in the ramification filtration ⓘ ramification filtration ⓘ upper numbering ramification groups ⓘ |
| context |
Galois theory of local fields
ⓘ
local class field theory ⓘ valuation theory ⓘ |
| contrastsWith | non-abelian extensions where upper ramification breaks may be non-integral ⓘ |
| field |
algebraic number theory
ⓘ
local field theory ⓘ ramification theory ⓘ |
| holdsFor | finite abelian Galois extensions of non-archimedean local fields ⓘ |
| implies |
Artin conductors of characters of abelian local Galois groups are integers
ⓘ
Swan conductors of characters of abelian local Galois groups are integers ⓘ breaks in the upper ramification filtration of abelian extensions are integers ⓘ ramification breaks of abelian extensions in upper numbering are not fractional ⓘ |
| involves |
Galois group filtration by higher ramification groups
ⓘ
complete discretely valued fields with finite residue field ⓘ discrete valuation fields ⓘ |
| namedAfter |
Cahit Arf
ⓘ
Helmut Hasse ⓘ |
| relatedTo |
Artin conductor
ⓘ
Herbrand function ⓘ Swan conductor ⓘ ramification breaks ⓘ |
| relates | upper numbering and lower numbering of ramification groups ⓘ |
| status |
classical theorem in local field theory
ⓘ
fundamental result in algebraic number theory ⓘ |
| typeOf | integrality theorem in ramification theory ⓘ |
| usedIn |
analysis of wild ramification
ⓘ
computation of discriminants of abelian extensions ⓘ local class field theory reciprocity ⓘ study of conductors of Galois representations ⓘ study of local L-factors ⓘ |
| uses |
lower numbering of ramification groups
ⓘ
upper numbering of ramification groups ⓘ |
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: Hasse–Arf theorem Description of subject: The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.