Hasse norm theorem
E213039
The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
All labels observed (2)
How this entity was disambiguated
This entity first appeared as the object of triple T1862417 — 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 norm theorem Context triple: [Helmut Hasse, notableWork, Hasse norm theorem]
-
A.
Hasse principle
The Hasse principle is a concept in number theory stating that a Diophantine equation has a rational solution if and only if it has solutions in all completions of the rationals, such as the real numbers and p-adic numbers.
-
B.
Hasse–Arf theorem
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.
-
C.
Hasse invariant
The Hasse invariant is an arithmetic invariant in number theory and algebraic geometry that classifies structures such as quadratic forms or elliptic curves over local and global fields, playing a key role in local-global principles.
-
D.
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.
-
E.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
- 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 norm theorem Target entity description: The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
-
A.
Hasse principle
The Hasse principle is a concept in number theory stating that a Diophantine equation has a rational solution if and only if it has solutions in all completions of the rationals, such as the real numbers and p-adic numbers.
-
B.
Hasse–Arf theorem
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.
-
C.
Hasse invariant
The Hasse invariant is an arithmetic invariant in number theory and algebraic geometry that classifies structures such as quadratic forms or elliptic curves over local and global fields, playing a key role in local-global principles.
-
D.
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.
-
E.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
result in algebraic number theory
ⓘ
theorem ⓘ |
| appliesTo |
cyclic extensions of global fields
ⓘ
function fields of one variable over finite fields ⓘ global fields ⓘ number fields ⓘ |
| assumes |
K is a global field
ⓘ
L over K is a finite cyclic extension ⓘ |
| characterizes | norms from cyclic extensions ⓘ |
| concerns | elements that are everywhere local norms ⓘ |
| conclusion | local norm conditions are sufficient for global norm representation in cyclic extensions ⓘ |
| failsInGeneralFor | non-cyclic extensions ⓘ |
| field | algebraic number theory ⓘ |
| generalizes | Hasse principle for norms in cyclic extensions ⓘ |
| hasFormulationIn |
Galois cohomology
ⓘ
cohomological terms ⓘ idele-theoretic language ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| holdsFor | cyclic Galois extensions ⓘ |
| involves |
global class field theory
ⓘ
idele class groups ⓘ idele groups ⓘ local class field theory ⓘ local completions of global fields ⓘ norm map from an extension field to its base field ⓘ |
| namedAfter | Helmut Hasse ⓘ |
| provenBy | Helmut Hasse ⓘ |
| relatedTo |
Artin reciprocity law
ⓘ
Hasse principle ⓘ Herbrand quotient ⓘ Hilbert reciprocity law ⓘ
linked to:
Hilbert symbol
Shafarevich group of a torus ⓘ Tate cohomology ⓘ global reciprocity map ⓘ |
| relates | global norm conditions to local norm conditions ⓘ |
| states |
for a cyclic extension L/K of global fields, an element of K is a global norm from L if and only if it is a local norm at every place of K
ⓘ
linked to:
Hasse norm theorem
|
| usedIn |
analysis of weak approximation on norm varieties
ⓘ
arithmetic of algebraic tori ⓘ class field theory ⓘ computations of relative Brauer groups ⓘ description of norm groups in cyclic extensions ⓘ proofs of local-global principles ⓘ study of norm one tori ⓘ |
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 norm theorem Description of subject: The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
Hasse norm theorem
→
states
→
for a cyclic extension L/K of global fields, an element of K is a global norm from L if and only if it is a local norm at every place of K
ⓘ
linked to: Hasse norm theorem