Counting number of maximal ideals with given property? There's a small theorem in algebra that I've been toying with. Let $R$ be integrally closed in its quotient field, and $E$ a splitting extension of the quotient field, and $S$ be the integral closure of $R$ in $E$. Then for a maximal ideal $p$ of $R$, there are only finitely many maximal ideals of $S$ whose intersection with $R$ is precisely $p$.
I was trying to formulate a small example. Let $f(X)=X^3+X+1$, and $E\supseteq\mathbb{Q}$ a splitting extension for $f$. So taking $R=\mathbb{Z}$ in this case, let $S$ be the integral closure of $\mathbb{Z}$ in $E$. I choose say $(3)$ as a maximal ideal of $\mathbb{Z}$. So there must be finitely many maximal ideals $q$ of $S$ such that $q\cap\mathbb{Z}=(3)$.
This sparked my curiosity, if we know that there are only finitely many such maximal ideals $q$, how many are there actually, at least in this case? I couldn't think of a way to actually enumerate them all to be able to say "There are 9 such maximal ideals!," or something along those lines. Thank you kindly.

Let  $R$ be a noetherian ring with fraction field $K=Frac(R)$.
 Let $K\subset E$ be a finite separable extension field of dimension $[E:K]=n$ and let $S$ be the integral closure of $R$ in $E$: it is also a Dedekind ring. 
Now given a non-zero prime $\mathfrak p\subset R$, decompose the extended ideal  $\mathfrak p\cdot S\subset S$ as : 
$$\mathfrak p\cdot S=\Pi \mathfrak P^{e_\mathfrak P}$$
(you can do that, since $S$ is Dedekind)
You then have the fundamental formula 
$$n=\Sigma  e_{\mathfrak P } \cdot f_ {\mathfrak P}   $$
where $f_ {\mathfrak P}=[S/\mathfrak P:R/\mathfrak p]$.
The $\mathfrak P$'s in the above formulae are exactly the primes of $S$ over $\mathfrak p$ and there are thus at most $n$ of them. This is the bound you were looking for.  
In your example, the splitting field $E$ of $X^3+X+1$ over $\mathbb Q$ has dimension 6 over $\mathbb Q$ and so you have at most 6 prime ideals over the prime ideal $\mathfrak p = (3)$ .  
Bibliography J.-P. Serre, Local Fields, Chapter 1, §4.
