Definition:Primitive Polynomial

Definition

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

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

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

$(1): \quad f \in \Z \left[{X}\right]$

$(2): \quad \operatorname{cont} \left({f}\right) = 1$

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

That is:

$(1): \quad$ All the coefficients of $f$ are integers

$(2): \quad$ The greatest common divisor of the coefficients of $f$ is equal to $1$.

Then $f$ is described as primitive.

Sources

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