TITLE: Ideal Class Group is finite
QUESTION [6 upvotes]: I'm having trouble understanding the proof given in class that the ideal class group of finite algebraic extension is finite.
Here it is (with my questions):
First, a lemma: Any class of ideals in $Cl(K)$ contains a representative $L$ for which $\mathcal{O}_K \subset L$ and $[L:\mathcal{O}_K]<(\frac{4}{\pi})^{r_2}\cdot\frac{n!}{n^n}\cdot\sqrt{|Disc(\mathcal{O})|}$.
Proof:

I'm already confused because the proof does not start by taking a certain class in $Cl(K)$ so I don't understand its structure.

Take a lattice $M$ in $K$ such that $\mathcal{O}_K=\mathcal{O}(M)$ where
$\mathcal{O}(M)=\{x\in K \mid xM \subset M\}$.

Why does such $M$ exist?

W.L.O.G we have $M \subset \mathcal{O}_K$ (I'm fine with this step because we can multiply $M$ by some scalar to make it so).
By Minkowski's theorem, we have $|Nm_{K/\mathbb{Q}}|\leq (\frac{4}{\pi})^{r_2}\cdot\frac{n!}{n^n}\cdot\sqrt{|Disc(\mathcal{O}_K)|}\cdot[\mathcal{O}_K:M]$.

Where did the $[\mathcal{O}_K:M]$ factor come from? I don't see it in Minkowski's theorem.

So $\alpha\mathcal{O}_K \subset M$ because $M$ is both a lattice and a ring.

I don't understand this implication.

Take $L=\alpha^{-1}M$.

Since we didn't start with a specific class, I don't understand what this $L$ has to do with anything.

We now have $[\alpha^{-1} M:\mathcal{O}]\leq[\alpha^{-1}M:M]=|Nm_{K/\mathbb{Q}}(\alpha)|\leq (\frac{4}{\pi})^{r_2}\cdot\frac{n!}{n^n}\cdot\sqrt{|Disc(\mathcal{O})|}$.
That ends the proof of the lemma. The proof of the theorem itself is then more clear.

REPLY [2 votes]: Usually, the proof goes in the following way. 
Let $M'$ be a fractionnal ideal. Write $M'=aM$ with $M\subseteq O_K$. By Minkowski, there exists $\alpha\in M$ non-zero such that
$$ \mathrm{Nm}_{K/\mathbb Q}(\alpha) \le C [O_K: M]$$ 
where $C=(4/\pi)^{r_2}(n!/n^n)\sqrt{|Disc(O_K)|}$. Let $L=\alpha^{-1}M$. Then 
$$[L:O_K]=[\alpha^{-1}M: O_K]=[M: \alpha O_K]=\mathrm{Nm}_{K/\mathbb Q}(\alpha)/[O_K: M]\le C.$$
By construction, $L=(a\alpha)^{-1}M'$ is a representative of the class of $M'$.