Primitive Polynomials over Z

# Primitive Polynomials over Z

 Definition: Let $f \in \mathbb{Z}[x]$ with $f(x) = a_0 + a_1x + ... + a_nx^n$. Then $f$ is said to be a Primitive Polynomial over $\mathbb{Z}$ if $\gcd (a_0, a_1, ..., a_n) = 1$.

For example, the following polynomial is a primitive polynomial over $\mathbb{Z}$:

(1)
\begin{align} \quad f(x) = 1 + 2x + 3x^2 + 5x^3 \end{align}

This is because $\gcd (1, 2, 3, 5) = 1$.

However, the next polynomial below is not a primitive polynomial over $\mathbb{Z}$:

(2)
\begin{align} \quad g(x) = 4 + 4x -2x^2 \end{align}

This is because $\gcd (4, 4, -2) = 2$.