Gauss's lemma (polynomials)

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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). Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. Gauss's lemma first appeared as Article 42 in his Disquisitiones Arithmeticae as the statement that if P and Q are monic polynomials with rational coefficients, not all of them integers, then the coefficients of their product PQ also cannot all be integers. In modern form, Gauss's lemma asserts that the product of two primitive polynomials is primitive. (A polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients.) A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it...

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