Determine maximal ideal in $(\mathbb{Z}[x], \langle f(x)\rangle)$

Question

I want to know

1. whether $(\mathbb{Z}[x], \langle x^5-4x+22\rangle)$ is a maximal ideal or not.

What I know is $\mathbb{Z}[x]$ has prime ideal for $(p)$, prime number and $(x)$ and maximal ideal has form like $(p,x)$. Above I write it down for special case, but for general case

2. How to determine $(\mathbb{Z}[x], \langle f(x)\rangle)$ is maximal ideal or not?

Answer 1

Theorem. $(f(x))$ is not a maximal ideal in $\mathbb Z[x]$ for any $f\in \mathbb Z[x]$.

Proof. We assume $f(x)$ is not constant. There must be a value such that $f(a)\neq\pm 1$. Notice that $((f(x))$ is contained in the ideal of polynomials $p$ such that $f(a)\mid p(a)$. To see the containment is proper notice that the constant polynomial with value $f(a)$ is not in $(f(x))$.

The case in which $f(x)$ is constant is also easy by taking the ideal of polynomials such that $f(0)$ divides $p(0)$.