Chevalley–Warning theorem

E559864

result in number theory theorem

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.

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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 NERFINISHED ⓘ
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 NERFINISHED ⓘ
implies	existence of nontrivial solutions under degree conditions ⓘ
involves	congruences modulo a prime ⓘ counting solutions modulo p ⓘ
namedAfter	Claude Chevalley NERFINISHED ⓘ Ernst Warning NERFINISHED ⓘ
originallyProvedBy	Claude Chevalley NERFINISHED ⓘ Ernst Warning NERFINISHED ⓘ
relatedTo	Ax–Katz theorem NERFINISHED ⓘ Hasse principle NERFINISHED ⓘ Weil conjectures NERFINISHED ⓘ 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 ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Claude Chevalley → knownFor → Chevalley–Warning theorem ⓘ