Gauss's Lemma states that the product of two primitive polynomials, meaning polynomials with integer coefficients whose coefficients share no common factor, is itself primitive, from which it follows that a polynomial with integer coefficients that factors over the rational numbers already factors over the integers. Named for Carl Friedrich Gauss, it is a basic tool for testing polynomials for irreducibility over the rationals.
Facts
StatementThe product of two primitive polynomials over the integers is itself primitive; equivalently, a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. 1 Classification
Statement FormCharacterization Theorem 1 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
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)
Proved By
Sources
1. Gauss's Lemma (Polynomials) (Wikipedia)
Wikimedia Foundationlead section, modern-form paragraphQuote, lead section, modern-form paragraph
In modern form, Gauss's lemma asserts that the product of two primitive polynomials is primitive.
2. Disquisitiones Arithmeticae (Wikipedia)
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written in Latin by Carl Friedrich Gauss in 1798, when Gauss was 21, and published in 1801, when he was 24
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