# Definition:Content of Polynomial

## Contents

## 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 field of quotients 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

- Definition:Primitive Polynomial (Ring Theory): A polynomial $f$ is
**primitive**if $\cont f = 1$.

- Results about
**Content of Polynomial**can be found here.