I am trying to solve the following problem from Artin:
Every maximal ideal $\mathbb{Z}[x]$ is of the form $(p,f)$ where $p$ is a prime integer and $f$ is a primitive polynomial that is irreducible modulo $p$.
My question is why do we need $f$ to be primitive? I have found this to be true without the assumption that $f$ is primitive.
It doesn't say that $f$ must be primitive, it says that $f$ can be chosen to be primitive.
For example, $(-2x+2,3)=(x+2,3)$. Notice $-2x+2$ is not primitive in $\Bbb Z[x]$, but $x+2$ is. This ideal is maximal in $\Bbb Z[x]$ whether you write $-2x+2$ or $x+2$.