# Extending a principal prime of $\mathbb{Z}[X]$ to a maximal one

Maximal ideals of $\mathbb{Z}[X]$ are of the form $(p, f(X))$, where $f$ reduces to an irreducible polynomial mod $p$. Now, given a principal prime ideal $\mathfrak{p}$ of $\mathbb{Z}[X]$, I would like to extend it to a maximal ideal.

If $\mathfrak{p}=(p)$ for some prime number $p$, this is easy. Just adjoin a polynomial which is irreducible mod $p$. In the other case, when $\mathfrak{p}=(f)$ for some $f$ irreducible in $\mathbb{Q}[X]$, I am having some trouble.

If $f$ is also irreducible modulo some prime, we can simply adjoin that prime to get a maximal ideal. But my problem lies in polynomials which are irreducible in $\mathbb{Q}[X]$ but reducible modulo every prime. A simple example is $X^4+1$.

How do I extend such primes to maximal ideals?

You could pick a prime number $p$, and one irreducible factor of $f$ modulo $p$.