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In order for the gcd to be exist, the polynomial must be nonzero; for reference, check p195 of Abstract Algebra (1st edition) by Gregory T. Lee Tags: Visual edit Mobile edit Mobile web edit |
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[[Gauss's lemma (polynomials)|Gauss's lemma for polynomials]] states that the product of primitive polynomials (with coefficients in the same unique factorization ___domain) also is primitive. This implies that the content and the primitive part of the product of two polynomials are, respectively, the product of the contents and the product of the primitive parts.
As the computation of greatest common divisors is generally much easier than [[polynomial factorization]], the first step of a polynomial factorization algorithm is generally the computation of its primitive part–content factorization (see {{slink|Factorization of polynomials|Primitive part–content factorization}}). Then the factorization problem is reduced to
Content and primitive part may be generalized to polynomials over the [[rational number]]s, and, more generally, to polynomials over the [[field of fractions]] of a unique factorization ___domain. This makes essentially equivalent the problems of computing [[polynomial greatest common divisor|greatest common divisors]] and factorization of polynomials over the integers and of polynomials over the rational numbers.
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