As the title states, I'm curious if the ideal $I=(X^4+4X^3+6,X-1)$ is prime/maximal in $\mathbb{Z}[X]$.
I think that this ideal is not prime/maximal in $\mathbb{Z}[X]$, because the prime ideals in $\mathbb{Z}[X]$ are either $0, (p), (f), (p,f)$ with $p$ being a prime number and $f$ being irreducible in $\mathbb{Z}[X]$. But maybe we can write $I$ as being equal to $(p,f)$. I guess I should also ask if this ideal is principle to also check if we can write $I=(p)$ or $I=(f)$.
Any help is appreciated
