I'm trying to understand the Zariski topology on $\text{Spec}(\mathbb{Z})$. I've just learned about this new concept and I wanted to compute this topology for a more concrete example to see how it looks but I dont know how to start.
For now I have that $\text{Spec}(\mathbb{Z})= \{(p); \text{p prime} \}\cup \{0\}$ since $\mathbb{Z}$ is principal. Then I don't know how to compute $\text{V}((n))$ for $n\in \mathbb{Z}$ but I don't see how, can someone help me ?
If you want to to compute $\text{V}((n))$ you can consider the prime decomposition of $n$ : $n=p_1^{\alpha_1}\cdot ... \cdot p_r^{\alpha_r}$, then you have $$\text{V}((n))=\{(p) ; \text{p prime and }(n)\subset (p)\}=\{(p) ; \text{p prime and }p \mid n \} = \{(p_1),...,(p_n)\}$$
Does that helps you visualizing what the topology looks like ?