Primitive polynomial

Let $R$ be an integral domain and let $f\in R[x]$ be nonzero. The polynomial $f$ is primitive if its content is the whole ring, i.e. $c(f)=R$ (equivalently, the coefficients generate the unit ideal).

Primitivity formalizes “no nontrivial common factor in the coefficients.” It is the hypothesis in Gauss’s lemma , which links irreducibility in $R[x]$ to irreducibility over the fraction field. In a field, every nonzero polynomial is primitive because any nonzero coefficient is a unit .

Examples:

• In $\mathbb{Z}[x]$, $2x+3$ is primitive since $(2,3)=\mathbb{Z}$.
• In $\mathbb{Z}[x]$, $x^2-5x+10$ is primitive.
• In $\mathbb{Z}[x]$, $6x+9$ is not primitive since its content is $(3)$.