The notes I read gives following technique to factor an ideal in a number field without explanation. Can anyone explain how this technique works?
To factor the ideal $(2)$ in $\mathbb{Z}[\sqrt{-5}]$, the idea is to factor $x^2+5$ mod $2$. The result is $x^2+5 \equiv (x+1)^2$ mod $2$. Identify $x$ with $\sqrt{-5}$, then we get $P=(2,1+\sqrt{-5})$. Then $(2)=P^2$.
Why this technique works for all $\mathbb{Z}[\alpha]$ if the number field is $\mathbb{Q}[\alpha]$?
Theorem. Let $K={\Bbb Q}(\alpha)$ be a number field, $\alpha\in{\cal O}_K$, and $f\in\Bbb Z[x]$ the minimal polynomial of the integer $\alpha$. For any $p$ not dividing $[{\cal O}_K:\Bbb Z[\alpha]]$ the shape of $f(x)$s factorization into irreducibles mod $p$ and the shape of $p{\cal O}_K$'s factorization into prime ideals in ${\cal O}_K$ are identical; furthermore there is a correspondence between them: if $\Pi(x)$ is a representative in $\Bbb Z[x]$ of one of the irreducible factors $\pi(x)$ of $f(x)$ mod $p$ then $(\Pi(\alpha),p)$ is the prime ideal in ${\cal O}_K$ corresponding to $\pi(x)$.
I believe this theorem is attributed to Dedekind. See this blurb of KCd and this MSE thread.
