Chevalley–Warning theorem
E559864
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Chevalley–Warning theorem canonical | 2 |
| Warning’s second theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5970292 — 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: Chevalley–Warning theorem Context triple: [Claude Chevalley, knownFor, Chevalley–Warning theorem]
-
A.
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.
-
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.
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.
-
D.
Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
-
E.
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.
- 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: Chevalley–Warning theorem Target entity description: The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
-
A.
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.
-
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.
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.
-
D.
Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
-
E.
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.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
result in number theory
ⓘ
theorem ⓘ |
| appearsIn |
texts on algebraic geometry over finite fields
ⓘ
texts on algebraic number theory ⓘ texts on arithmetic geometry ⓘ |
| appliesTo | finite fields of prime power order ⓘ |
| area |
algebraic number theory
ⓘ
arithmetic geometry ⓘ |
| assumes |
coefficients in a finite field
ⓘ
finite field of characteristic p ⓘ |
| concerns |
polynomial equations over finite fields
ⓘ
solutions in finite fields ⓘ systems of polynomial equations ⓘ |
| field |
algebraic geometry
ⓘ
number theory ⓘ |
| generalizedBy | Ax–Katz theorem ⓘ |
| givesConditionOn |
degrees of polynomials
ⓘ
number of variables ⓘ |
| guarantees |
existence of common zeros
ⓘ
existence of nontrivial common solutions ⓘ |
| hasConsequence |
divisibility of number of solutions by the characteristic
ⓘ
lower bounds on number of solutions ⓘ |
| hasProofTechnique |
combinatorial counting argument
ⓘ
use of polynomial identities over finite fields ⓘ |
| hasVariant |
Warning’s second theorem
ⓘ
linked to:
Chevalley–Warning theorem
|
| implies | existence of nontrivial solutions under degree conditions ⓘ |
| involves |
congruences modulo a prime
ⓘ
counting solutions modulo p ⓘ |
| namedAfter |
Claude Chevalley
ⓘ
Ernst Warning ⓘ |
| originallyProvedBy |
Claude Chevalley
ⓘ
Ernst Warning ⓘ |
| relatedTo |
Ax–Katz theorem
ⓘ
Hasse principle ⓘ Weil conjectures ⓘ local–global principles ⓘ |
| statedInTermsOf |
number of variables in the system
ⓘ
sum of degrees of polynomials ⓘ |
| topicOf | research in finite field theory ⓘ |
| usedIn |
applications to Diophantine equations over finite fields
ⓘ
counting rational points on varieties over finite fields ⓘ proofs in additive combinatorics ⓘ |
| uses |
combinatorial methods
ⓘ
properties of finite fields ⓘ |
| yearProved | 1935 ⓘ |
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: Chevalley–Warning theorem Description of subject: The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Chevalley–Warning theorem