Understanding proof of “The ring of integers of a number field is a Dedekind domain”

Let K be a number field and $$O_K$$ its ring of integers.

Trying to understand proof of "The ring of integers of a number field is a Dedekind domain", in the part:

Every element of the finitely generated abelian group $$O_K /p$$ is killed by $$a_0$$ , so $$O_K /p$$ is a finite set. What does it mean?

Thank you.

• You quoted two sentences. What exactly do you not understand? – Bernard Jul 22 '19 at 20:16
• Please check the sentence in bold, how does $a_0$ kill the elements in $O_K/p$ and why $O_K/p$ is a finite set.Thank you. – PerelMan Jul 22 '19 at 20:18

$$a_0$$ is a nonzero integer, and $$a_0(\mathcal O_K/\mathfrak p)=0$$, that is $$\mathfrak{p}\supseteq|a_0|\mathcal{O}_K$$ and so $$\mathcal O_K/\mathfrak p$$ is a quotient of $$\mathcal O_K/|a_0|\mathcal O_K$$. Therefore $$|\mathcal O_K/\mathfrak p|\le|\mathcal O_K/|a_0|\mathcal O_K|$$. But $$|\mathcal O_K/|a_0|\mathcal O_K|=|a_0|^n<\infty$$, where $$n=|K:\Bbb Q|$$ since as an Abelian group, $$\mathcal O_K$$ is free of rank $$n$$.
• Thank you. I don't know if $|\mathcal O_K/|a_0|\mathcal O_K|=|a_0|^n$ is always true? how do we get that result? Thanks – PerelMan Jul 22 '19 at 23:08
$$a_0$$ kills the elements of $$\mathcal O_K/\mathfrak p$$ simply because the author has just explained why $$a_0 \in \mathfrak p$$.
On the other hand, $$\mathcal O_K/\mathfrak p$$ is a finitely generated abelian group, and the above remark shows it is torsion, which implies finiteness, by the Structure theorem for finitely generated abelian groups.