Hilbert’s irreducibility theorem

E43322

mathematical theorem theorem in algebraic geometry theorem in number theory

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.

Aliases (4)

Statements (48)

Predicate	Object
instanceOf	mathematical theorem → theorem in algebraic geometry → theorem in number theory →
appearsIn	Hilbert’s work on algebraic number fields →
appliesTo	polynomials with coefficients in number fields → polynomials with parameterized coefficients →
characterizes	Hilbertian fields →
concerns	irreducibility of specialized polynomials → specialization of parameters in polynomials →
context	polynomials irreducible over a function field remaining irreducible after specialization → rational function fields over number fields →
ensures	many polynomial equations with parameters remain irreducible for infinitely many specializations of the parameters →
field	algebra → algebraic geometry → number theory →
formalizedIn	the language of Hilbertian fields and thin sets →
generalizes	classical results on irreducibility of polynomials over the rationals →
hasConsequence	existence of infinitely many specializations with prescribed local behavior → rational points on certain varieties are Zariski dense →
hasVersion	Hilbert’s irreducibility theorem → surface form: "arithmetic Hilbert irreducibility theorem" Hilbert’s irreducibility theorem → surface form: "geometric Hilbert irreducibility theorem"
historicalPeriod	late 19th century mathematics →
holdsOver	number fields → the rational numbers →
implies	existence of infinitely many specializations preserving Galois group in many cases → existence of infinitely many specializations preserving irreducibility →
influenced	development of modern inverse Galois theory → techniques in arithmetic geometry →
involves	Zariski open subsets of affine space → thin sets in number theory →
namedAfter	David Hilbert →
provenBy	David Hilbert →
relatedTo	Chebotarev density theorem → Hilbert’s irreducibility theorem → surface form: "Hilbertian fields" Hilbert’s Nullstellensatz →
status	fundamental result in number theory and algebraic geometry →
toolFor	eliminating parameters in Diophantine problems → reducing problems over function fields to problems over number fields →
usedFor	producing infinitely many linearly disjoint field extensions → realization of finite groups as Galois groups over number fields → specialization of covers of the projective line →
usedIn	Diophantine geometry → Hilbert’s irreducibility theorem → surface form: "Hilbert irreducibility method" arithmetic geometry → construction of Galois extensions of number fields → inverse Galois theory → proofs of existence of field extensions with prescribed Galois group → specialization arguments in algebraic geometry →

Referenced by (5)

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

Hilbert’s irreducibility theorem → hasVersion → Hilbert’s irreducibility theorem →

this entity surface form: "geometric Hilbert irreducibility theorem"

Hilbert’s irreducibility theorem → hasVersion → Hilbert’s irreducibility theorem →

this entity surface form: "arithmetic Hilbert irreducibility theorem"

David Hilbert → notableWork → Hilbert’s irreducibility theorem →

Hilbert’s irreducibility theorem → relatedTo → Hilbert’s irreducibility theorem →

this entity surface form: "Hilbertian fields"

Hilbert’s irreducibility theorem → usedIn → Hilbert’s irreducibility theorem →

this entity surface form: "Hilbert irreducibility method"