Primitive part and content: Difference between revisions

Revision as of 22:14, 10 July 2021

Revision as of 22:17, 10 July 2021

Line 3:

In [[algebra]], the '''content''' of a [[polynomial]] with integer coefficients (or, more generally, with coefficients in a [[unique factorization ___domain]]) is the [[greatest common divisor]] 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 [[up to]] the multiplication of the content by a [[unit (ring theory)|unit]] of the [[ring (mathematics)|ring]] of the coefficients (and the multiplication of the primitive part by the [[multiplicative inverse|inverse]] of the unit).

A polynomial is ''[[Primitive polynomial (ring theory)|'primitive]]''' if its content equals 1. Thus the primitive part of a polynomial is a primitive polynomial.

[[Gauss's lemma (polynomial)|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.