Example of an ideal which is not principal in the ring $\mathbb{Z} [x]$

Question

Give an example of an ideal in the ring $\mathbb{Z} [x]$ is not principal. What kind of example would be the easiest?

Answer 1

Try the following one: $\;I=\langle 2,x\rangle\le\Bbb Z[x]\;$ .

Try first to characterize the elements of the ideal (take a good peek at the free coefficient of this

ideal's elements...), and now assume that $\;I=\langle f(x)\rangle\;,\;\;f(x)\in\Bbb Z[x]\;$ and get a contradiction.