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?
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 $\mathcal{O}_K$.