# Definition:Content of Polynomial

## Contents

## Definition

### Integer Polynomial

Let $f \in \Z \left[{X}\right]$ be a polynomial.

Then the **content** of $f$, $\operatorname{cont} \left({f}\right)$ is the greatest common divisor of the coefficients of $f$.

### Rational Polynomial

If $f \in \Q \left[{X}\right]$ then there is some $n \in \N$ such that $n f \in \Z \left[{X}\right]$.

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

- $\operatorname{cont} \left({f}\right) := \dfrac {\operatorname{cont} \left({n f}\right)} n$

### Polynomial over a GCD Domain

Let $D$ be a GCD domain.

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

Let $f \in K \left[{X}\right]$ be a polynomial.

Let $a \in D$ be such that $af \in D \left[{X}\right]$.

Let $d$ be the greatest common divisor of the coefficients of $af$.

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

- $\operatorname{cont} \left({f}\right) := \dfrac d a$

## General ring with unity

Let $R$ be a commutative ring with unity.

Let $f \in R[X]$ be a polynomial.

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

## Also denoted as

$\operatorname{cont} \left({f}\right)$ is also seen denoted $c_f$ or $\left\langle\hspace{-4mu}\left\langle{f}\right\rangle\hspace{-4mu}\right\rangle$.

## Also see

- Content of Polynomial over UFD is Well Defined
- Content of Polynomials is Multiplicative
- Definition:Primitive Polynomial: A polynomial $f$ is
**primitive**if $\operatorname{cont} \left({f}\right) = 1$.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): $\S 6.31$: Theorem $61$