Hilbert's Irreducibility Theorem, conceived by David Hilbert in 1892, is a result of number theory stating that for any finite set of polynomials in several variables with rational coefficients, each irreducible over the rational numbers, there exists a common way to substitute rational number values for a proper subset of the variables so that every one of the polynomials remains irreducible. The theorem is a foundational tool in number theory and in the inverse Galois problem, since it allows properties established generically, for indeterminate parameter values, to be specialized down to concrete rational number instances while preserving irreducibility.
Facts
StatementEvery finite set of irreducible polynomials in a finite number of variables with rational number coefficients admits a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Hilbert's irreducibility theorem, Wikipedia
Named After
Derived from the theorem's own name (unambiguous possessive-token match to exactly one live mathematician entity, w-bfill-g5-0924 browse backfill)
Sources
1. Hilbert's irreducibility theorem, Wikipedia
Statement section
every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible
Lead section
conceived by David Hilbert in 1892
In Branch: Number Theory, Lead sentence
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducibl
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