Primitive part and content

In <a href="/facts/Algebra/nMdqczjo">algebra</a>, the content of a nonzero <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> with <a href="/facts/Integer/PUwwrawx">integer</a> <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> (or, more generally, with coefficients in a <a href="/facts/Unique_factorization_domain/aYsX2swC">unique factorization domain</a>) is the <a href="/facts/Greatest_common_divisor/kNFvBuKl">greatest common divisor</a> of its coefficients. The primitive part of such a polynomial is the quotient of the polynomial by its content. Thus a polynomial is the product of its primitive part and its content, and this factorization is unique <a href="/facts/Up_to/ydz4ztzX">up to</a> the multiplication of the content by a <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">unit</a> of the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> of the coefficients (and the multiplication of the primitive part by the <a href="/facts/Multiplicative_inverse/yczdYyCw">inverse</a> of the unit).
A polynomial is primitive if its content equals 1. Thus the primitive part of a polynomial is a primitive polynomial.
<a href="/facts/Gauss%27s_lemma_(polynomials)/gATpSMek">Gauss's lemma for polynomials</a> 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 <a href="/facts/Polynomial_factorization/1S9HRKcI">polynomial factorization</a>, the first step of a polynomial factorization algorithm is generally the computation of its primitive part–content factorization (see <a href="/facts/Factorization_of_polynomials/1S9HRKcI">Factorization of polynomials § Primitive part–content factorization</a>). Then the factorization problem is reduced to factorize separately the content and the primitive part.
Content and primitive part may be generalized to polynomials over the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>, and, more generally, to polynomials over the <a href="/facts/Field_of_fractions/cmlJrDSS">field of fractions</a> of a unique factorization domain. This makes essentially equivalent the problems of computing <a href="/facts/Polynomial_greatest_common_divisor/4c3dUbJr">greatest common divisors</a> and factorization of polynomials over the integers and of polynomials over the rational numbers.

Primitive part and content open-in-new

Primitive part and content