Hasse principle
E207311
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hasse principle canonical | 4 |
| Hasse–Minkowski theorem | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T1862412 — 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 principle Context triple: [Helmut Hasse, notableWork, Hasse principle]
-
A.
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.
-
B.
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.
-
C.
Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
-
D.
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.
-
E.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
- 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 principle Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
-
D.
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.
-
E.
Fermat's theorem on sums of two squares
Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in number theory
ⓘ
local-global principle ⓘ mathematical principle ⓘ |
| appliesTo |
certain conics over Q
ⓘ
norm forms in cyclic extensions in some cases ⓘ quadratic equations over Q ⓘ quadratic forms ⓘ |
| assumes |
existence of local solutions at every place of Q
ⓘ
knowledge of completions of Q ⓘ |
| completion |
p-adic numbers
ⓘ
surface form:
p-adic numbers Q_p
real numbers R ⓘ |
| conclusion | existence of a rational point when the principle holds ⓘ |
| connectedTo |
class field theory in some applications
ⓘ
cohomological obstructions to rational points ⓘ |
| describes | relationship between local and global solvability of Diophantine equations ⓘ |
| domain | Diophantine equations ⓘ |
| failsFor |
some cubic curves
ⓘ
some curves of genus at least 1 ⓘ some higher-degree Diophantine equations ⓘ |
| field | number theory ⓘ |
| formalSetting | algebraic varieties over Q ⓘ |
| globalCondition | existence of a rational solution ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| holdsFor | quadratic forms over Q ⓘ |
| influencedBy | work of Helmut Hasse on quadratic forms ⓘ |
| involves |
archimedean and non-archimedean completions
ⓘ
places of Q ⓘ |
| localCondition | existence of solutions over all completions of Q ⓘ |
| motivation | to infer global information from local data ⓘ |
| namedAfter | Helmut Hasse ⓘ |
| quantifiesOver |
solutions in Q
ⓘ
solutions in Q_p for all primes p ⓘ solutions in R ⓘ |
| relatedConcept |
Brauer–Manin obstruction
ⓘ
Hasse principle self-linksurface differs ⓘ
surface form:
Hasse–Minkowski theorem
global solvability ⓘ local solvability ⓘ rational points on algebraic varieties ⓘ weak approximation ⓘ |
| statement | a Diophantine equation has a rational solution if and only if it has solutions over all completions of the rationals ⓘ |
| status | valid for some classes of equations and invalid for others ⓘ |
| typeOf | local-to-global criterion ⓘ |
| typicalFormulation | X(Q) is nonempty if and only if X(R) and X(Q_p) are nonempty for all primes p ⓘ |
| usedIn |
algebraic number theory
ⓘ
arithmetic geometry ⓘ rational points theory ⓘ |
| uses | completions of the 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: Hasse principle Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hasse–Minkowski theorem
this entity surface form:
Hasse–Minkowski theorem
this entity surface form:
Hasse–Minkowski theorem