Dedekind Zolotarev factorization

(Dedekind, Zolotarev) If \(\theta\) is an integral primitive element of \(K/\mathbb{Q}\) and a rational prime \(p\) does not divide \([\mathcal{O}_{K}:\mathbb{Z}[\theta] ]\), then \(p\)’s factorization in \(\mathcal{O}_{K}\) is of the same shape as that of \(\theta\)’s minimal polynomial \(f(x)\) in \(\mathbb{F}_{p}(X)\). That is, there is a natural bijection between primes \(\mathcal{P}|p\) and irreducibles \(\pi | f\) in \(\mathbb{F}_{p}(X)\), and the ramification and residue data match up.

Finally, \(\mathcal{P}= \langle p,\Pi(\theta)\rangle\) for any \(\Pi(X) \in \mathbb{Z}(X)\) such that \(\Pi(X) \equiv_{p} \pi(X)\).