Ideal in $\mathbb Z[x]$ which is not two-generated I can prove that the ideal $(4, 2x, x^2)$ in $\mathbb Z[x]$ is not principal. But I failed to prove that this cannot be generated by two elements.
It's really difficult for me. Would you give me a hint.

Note that $(4, 2x, x^2)=(2,x)^2$. Set $J=(2,x)$. We have $\mathbb Z[x]/J\simeq\mathbb F_2$ (the field with two elements). Show that $4, 2x, x^2$ are linearly independent in $J^2/J^3$ over $\mathbb F_2$, so $\dim_{\mathbb F_2}J^2/J^3\ge3$. If $J^2$ is two-generated, then the quotient $J^2/J^3$ is also two-generated, a contradiction.
