Why is $(2,x)$ non-principal in $\Bbb Z[x]$?

Question

Why is $(2,x)$ non-principal in $\Bbb Z[x]$?

Apparently this is the case, I just read it on wiki, as a counter example to $\Bbb Z[x]$ being a PID.

What is $x$ here? I mean $2$ can surely generate $x$ right?

Answer 1

If it were principal, generated by a polynomial $f$, this polynomial would have to divide $2$, hence be a constant (because over an integral domain, $\,\deg fg=\deg f+\deg g$) which is either a unit, or associated with $2$.

However, it can't be a unit, since $(2,x)$ is a maximal ideal in $\mathbf Z[x]$. Indeed, $$\mathbf Z[x]/(2,x)\simeq\mathbf Z/2\mathbf Z $$ is a field.

Nor can it be $2$ since $$\mathbf Z[x]/(2)\simeq(\mathbf Z/2\mathbf Z)[x] $$ is not a field, which proves $(2)$ is not a maximal ideal in $\mathbf Z[x]$.