Definition:Primitive Polynomial (Ring Theory)

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

 * Product of Primitive Polynomials is Primitive
 * Definition:Primitive Part of Polynomial