# Definition:Primitive Polynomial

## Contents

## Definition

Let $\Q \left[{X}\right]$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$.

Let $f \in \Q \left[{X}\right]$ be such that:

- $\operatorname{cont} \left({f}\right) = 1$

where $\operatorname{cont} \left({f}\right)$ is the content of $f$.

That is:

- The greatest common divisor of the coefficients of $f$ is equal to $1$.

Then $f$ is described as **primitive**.

## Also defined as

From Polynomial has Integer Coefficients iff Content is Integer it follows that, if $f$ is a **primitive polynomial**, then:

- $f \in \Z \left[{X}\right]$

where $\Z \left[{X}\right]$ is the ring of polynomial forms over the integral domain of integers in the indeterminate $X$.

Hence this definition can often be found stated with the additional condition that the coefficients of $f$ are integers, but it should be noted that this condition is in fact superfluous.

## Also see

## Sources

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