Prime ideals of the ring of integers of an algebraic number field I am working on a problem that has a completely different point and I didn't work with algebraic number fields much before, so I was wondering if someone could point me in the right direction for this:
How can prime ideals of the ring of integers of an algebraic number field be characterised? Since the ring is Dedekind they are all maximal, but what are their generators?
 A: Let $K$ be an algebraic number field, and let $\mathcal{O}_K$ be its ring of integers. $K$ contains $\mathbb{Q}$ and $\mathcal{O}_K$ contains $\mathbb{Z}$; for any prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$, $\mathfrak{p}\cap \mathbb{Z}$ is a prime ideal of $\mathbb{Z}$, which therefore has the form $(p)$ for a prime integer $p$. The prime $\mathfrak{p}$ of $\mathcal{O}_K$ is said to "lie over" $(p)$.
So, all the prime ideals of $\mathcal{O}_K$ are lying over the primes of $\mathbb{Z}$. The question is then how to find the primes that are lying over a given $(p)$ explicitly. For all but a finite number of primes $p$, this can be done as follows:
Find an $\alpha\in\mathcal{O}_K$ that generates $K$ over $\mathbb{Q}$ as a field. Then find a minimal polynomial $f$ for $\alpha$ over $\mathbb{Q}$. $f$ is a monic integer polynomial because of how $\alpha$ was chosen. Now consider $f$'s image mod $p$, and factor it into irreducible factors. If $g$ is any irreducible factor of $f$ mod $p$, then the ideal generated by $p$ and $g(\alpha)$ is a prime ideal of $\mathcal{O}_K$ lying over $p$. Thus the primes over $p$ correspond in a lovely way with irreducible factors of $f$ mod $p$.
This is guaranteed to work unless $p$ is one of the primes dividing the index of the ring $\mathbb{Z}[\alpha]$ as an additive subgroup of $\mathcal{O}_K$, of which there are only finitely many and in favorable cases there may be none (i.e. when you can find $\alpha$ such that $\mathcal{O}_K = \mathbb{Z}[\alpha]$).
Even in the cases where this algorithm doesn't work, if you have an explicit description of $\mathcal{O}_K$, you may still be able to find the primes lying over $p$ by finding the prime ideals in $\mathcal{O}_K/p\mathcal{O}_K$, which is a finite ring, and pulling them back to $\mathcal{O}_K$.
