How to prove this ideal is a prime ideal

Question

Let $I=\langle 3, f\rangle$ be an ideal in $\mathbb{Z}[X]$, where $f(x)=x^3+x^2+1$. How do I show that $I$ is a prime ideal?

I know that $I$ is not a principal ideal, using an argument similar to: Why is $(3,x^3-x^2+2x-1)$ not principal in $\mathbb{Z}[x]$?

I tried proving using definition, letting $ab\in I$ and trying to show that $a\in I$ or $b\in I$.

Let $ab\in I$. Then $ab=3g+fh$ for some $g,h\in\mathbb{Z}[X]$.

At this point, I have no idea what to do. I tried using the division algorithm to write $g=q_1f+r_1$, where $\deg r_1<3$, and $h=q_2f+r_2$, where $\deg r_2<3$.

Then $ab=3q_1f+3r_1+f^2q_2+fr_2$. I am not sure if this is the right track, or it just complicates things further.

Thanks for any help!

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I also had the idea of using: $I$ is a prime ideal iff $R/I$ is an integral domain, but I realised that it amounts to the same approach as above.

Answer 1

It's not a prime ideal. $(x-1)(x^2-x-1)\in I$, while $x-1\not\in I$ and $x^2-x-1\not\in I$.