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Showing the ring of integers is a Dedekind domain

Let $K$ be a number field and $O_K$ be the ring of integers of $K$. I want to show that $O_K$ is a Dedekind domain.
I saw this theorem in Wikipedia,
If $K$ is a number field and $O_K$ is the ring of integers of $K$, then
a) $O_K$ is a Dedekind domain.
b) $O_K$ is a UFD.
c) The ring of integers of a number field is a Dedekind domain.
I am not sure about how to prove this theorem.

A:

Let $L$ be the number field $\mathbb{Q}(\sqrt[4]{5})$. $L$ is a real quadratic extension of $\mathbb{Q}$ and $O_L$ is a Dedekind domain. Therefore every $\mathbb{Z}$-submodule of $O_L$ containing $1$ has a $\mathbb{Z}$-generator. Now $O_K$ contains $O_L$, therefore $O_K$ contains $1$, hence $O_K$ is a Dedekind domain.

A:

Let $A=\mathbb Z \cap O_K$ be the intersection of the ring of integers and the ring of integers. It’s a $\mathbb Z$-module containing $1$ and with non zero divisors. By Hilbert’s theorem 90, the group $A$ is free so it’s free as a $\mathbb Z$-module and this implies that it’s projective (by Nakayama’s lemma) so that $\text{fd}_{\mathbb Z}A=\text{fd}_{\mathbb Z}\mathbb Z=0$.

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