Primitive part and content: Difference between revisions

Revision as of 15:49, 26 May 2022

Revision as of 05:30, 5 February 2023

Line 25:

The '''content''' {{math|''c''(''P'')}} of a polynomial {{math|''P''}} with coefficients in {{math|''R''}} is the greatest common divisor of its coefficients, and, as such, is defined up to multiplication by a unit. The '''primitive part''' {{math|pp(''P'')}} of {{math|''P''}} is the quotient {{math|''P''/''c''(''P'')}} of {{math|''P''}} by its content; it is a polynomial with coefficients in {{math|''R''}}, which is unique up to multiplication by a unit. If the content is changed by multiplication by a unit {{math|''u''}}, then the primitive part must be changed by dividing it by the same unit, in order to keep the equality

:<math display="block">P = c(P) \operatorname{pp}(P),</math>

which is called the primitive-part-content factorization of {{math|''P''}}.

The main properties of the content and the primitive part are results of [[Gauss's lemma (polynomial)|Gauss's lemma]], which asserts that the product of two primitive polynomials is primitive, where a polynomial is primitive if 1 is the greatest common divisor of its coefficients. This implies:

*The content of a product of polynomials is the product of their contents: <math display="block">c(P_1 P_2) = c(P_1) c(P_2).</math>

*The primitive part of a product of polynomials is the product of their primitive parts: <math display="block"> \operatorname{pp}(P_1 P_2) = \operatorname{pp}(P_1) \operatorname{pp}(P_2).</math>

::<math>c(P_1P_2)=c(P_1)c(P_2).</math>

*The content of a greatest common divisor of polynomials is the greatest common divisor (in {{math|''R''}}) of their contents: <math display="block"> c(\operatorname{gcd}(P_1, P_2)) = \operatorname{gcd}(c(P_1), c(P_2)).</math>

*The primitive part of a product of polynomials is the product of their primitive parts:

*The primitive part of a greatest common divisor of polynomials is the greatest common divisor (in {{math|''R''}}) of their primitive parts:: <math display="block"> \operatorname{pp}(P_1P_2\operatorname{gcd}(P_1, P_2)) = \operatorname{gcd}(\operatorname{pp}(P_1), \operatorname{pp}(P_2)).</math>

*The content of a greatest common divisor of polynomials is the greatest common divisor (in {{math|''R''}}) of their contents:

::<math>c(\operatorname{gcd}(P_1, P_2))=\operatorname{gcd}(c(P_1), c(P_2)).</math>

*The primitive part of a greatest common divisor of polynomials is the greatest common divisor (in {{math|''R''}}) of their primitive parts:

::<math>\operatorname{pp}(\operatorname{gcd}(P_1, P_2))=\operatorname{gcd}(\operatorname{pp}(P_1), \operatorname{pp}(P_2)).</math>

*The complete [[factorization of polynomials|factorization]] of a polynomial over {{math|''R''}} is the product of the factorization (in {{math|''R''}}) of the content and of the factorization (in the polynomial ring) of the primitive part.