Let $K$ be a quadratic number field, and $\mathcal{O}_K$ the ring of integers of $K$.
The map $\pi: Spec(\mathcal{O}_K) \rightarrow Spec(\mathbb{Z})$ that sends a prime ideal $\mathbb{p}$ to $\mathbb{p} \cap \mathbb{Z}$ is induced since $\mathcal{O}_K$ contains $\mathbb{Z}$. And the fiber $\pi^{-1}$ of the prime ideal $(p)$ of $\mathbb{Z}$ is then understood as the decomposition of $(p)$ in $\mathcal{O}_K$.
We then obtain a geometric interpretation of how p factors in $\mathcal{O}_K$ using results obtained from Algebraic number theory.
I'm looking for hints to (major or minor!) results that can be proved regarding the behaviour of $(p)$ in $\mathcal{O}_K$, or other interesting aspects of $\mathcal{O}_K$ using "as much as possible" Algebraic geometry (at level of a first course in Schemes, using, say first seven chapters of Liu's "Algebraic Geometry and Arithmetic Curves").
I'd also appreciate a recommendation of a textbook or notes that discusses these ideas in details.