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.

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All labels observed (5)

Label	Occurrences
Hilbert irreducibility method	1
Hilbertian fields	1
Hilbert’s irreducibility theorem canonical	1
arithmetic Hilbert irreducibility theorem	1
geometric Hilbert irreducibility theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T326974 — 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: Hilbert’s irreducibility theorem
Context triple: [David Hilbert, notableWork, Hilbert’s irreducibility 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. 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.
C. Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
D. Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
E. Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
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: Hilbert’s irreducibility theorem
Target entity description: 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.

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. 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.
C. Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
D. Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
E. Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
F. None of above. chosen

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	arithmetic Hilbert irreducibility theorem ⓘ linked to: Hilbert’s irreducibility theorem geometric Hilbert irreducibility theorem ⓘ linked to: Hilbert’s 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 ⓘ Hilbertian fields ⓘ linked to: Hilbert’s irreducibility theorem 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 irreducibility method ⓘ linked to: Hilbert’s irreducibility theorem 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 ⓘ

How these facts were elicited

Referenced by (5)

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

David Hilbert → notableWork → Hilbert’s irreducibility theorem ⓘ

Hilbert’s irreducibility theorem → usedIn → Hilbert irreducibility method ⓘ

linked to: Hilbert’s irreducibility theorem

Hilbert’s irreducibility theorem → relatedTo → Hilbertian fields ⓘ

linked to: Hilbert’s irreducibility theorem

Hilbert’s irreducibility theorem → hasVersion → geometric Hilbert irreducibility theorem ⓘ

linked to: Hilbert’s irreducibility theorem

Hilbert’s irreducibility theorem → hasVersion → arithmetic Hilbert irreducibility theorem ⓘ

linked to: Hilbert’s irreducibility theorem

All labels observed (5)

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Statements (48)

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Referenced by (5)

How this entity was disambiguated

How these facts were elicited