I apologize if my abstract algebra is so shaky that I might have glossed over an answer to this problem.
Yesterday I suddenly got the motivation to do some number theory again - as a challenge, I wanted to see if I could figure out when and how rational primes could be factored in the ring of integers $\mathcal{O}$ of an arbitrary number field, inspired for example by $$17 = (1-\alpha)(1+\alpha)$$ where $\alpha = 4i$. To figure this out on my own, I selected $\mathcal{O}_{\mathbb{Q}(\alpha)}$ with a number satisfying $\alpha^3-\alpha-2=0$ as my ring of choice. After some searching around I found the neat Dedekind-Kummer theorem, which settles the factorization problem of ideals into prime ideals:
$$\langle p\rangle=\bigl\langle p,g_1(\alpha)\bigr\rangle ^{e_1}\,\bigl\langle p,g_2(\alpha)\bigl\rangle^{e_2}\cdots\bigl\langle p,g_k(\alpha)\bigl\rangle^{e_k}.$$
where the polynomial factors and exponents are corresponding to the factorization of a given polynomial $\bmod p$. At this point I thought I was done and tried to factor the prime $p = 11$ in $\mathcal{O}_{\mathbb{Q}(\alpha)}$ and got
$$\langle 11 \rangle = \langle 11,\alpha+8 \rangle \langle 11,\alpha^2+3\alpha+8 \rangle$$
but after reading up on some basic ring theory again, I realised that this isn't quite what I wanted to achieve and I started questioning myself if a concrete answer to this is even feasible. Since "primes" is often taken synonymously with "prime ideals" in this context, it is really hard to filter the search results for what I want to read up on here. My one/two questions are as follows:
Is "When is a given rational prime decomposable in a fixed arbitrary number field?" answerable? If this is possible, how can one algorithmically find the splitting of such a prime? How about my example with $\alpha^3-\alpha-2=0$?
Thank you.
