Primitive polynomial

Definition

Let $R$ be a commutative ring. A polynomial $f \in R[x]$ is primitive if every $r \in R$ that divides $f$ in $R[x]$ is a unit. Here $r$ divides $f$ when $r$ divides each coefficient of $f$.

Remarks

If $R$ is a unique factorisation domain, then $f$ is primitive if and only if its content is a unit. In $\mathbb{Z}[x]$, $3x^2 + 2x - 1$ is primitive and $6x^2 + 4x - 2$ is not. Every monic polynomial is primitive. The word primitive polynomial has another meaning over a finite field: a polynomial whose root is a primitive element.

Explanations

L1 human

A primitive polynomial has coefficients with no common factor. Over the integers, this means that the gcd of the coefficients is $1$.