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
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Sources
1. Gauss's Lemma (Polynomials) (Wikipedia)
Wikimedia Foundationlead section, modern-form paragraph
In modern form, Gauss's lemma asserts that the product of two primitive polynomials is primitive.
In Group: Polynomials and Elementary Algebra, Article text
In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a theorem about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic).
View the Source 2. Disquisitiones Arithmeticae (Wikipedia)
Wikimedia Foundationlead section, first sentenceQuote, lead section, first sentence
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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