Definition:Content of Polynomial

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Definition

Integer Polynomial

Let $f \in \Z \sqbrk X$ be a polynomial with integer coefficients.

Then the content of $f$, denoted $\cont f$, is the greatest common divisor of the coefficients of $f$.


Rational Polynomial

Let $f \in \Q \sqbrk X$ be a polynomial with rational coefficients.


The content of $f$ is defined as:

$\cont f := \dfrac {\cont {n f} } n$

where $n \in \N$ is such that $n f \in \Z \sqbrk X$.


Polynomial in GCD Domain

Let $D$ be a GCD domain.

Let $K$ be the quotient field of $D$.

Let $f \in K \sqbrk X$ be a polynomial.

Let $a \in D$ be such that $a f \in D \sqbrk X$.

Let $d$ be the greatest common divisor of the coefficients of $a f$.

Then we define the content of $f$ to be:

$\cont f := \dfrac d a$


Commutative Ring with Unity

Let $R$ be a commutative ring with unity.

Let $f \in R \sqbrk X$ be a polynomial.


The content of $f$ is the ideal generated by its coefficients.


Also denoted as

The content of a polynomial $f$ can be seen in the literature variously denoted as:

$\cont f$ (currently used on $\mathsf{Pr} \infty \mathsf{fWiki}$)
$c_f$
$\left\langle \! \left\langle {f} \right\rangle \! \right\rangle$


Also see

  • Results about Content of Polynomial can be found here.