some ring theory applied to holomorphic functions

Question

I'd like to know if my understanding of this business is correct.

Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromor­phic functions on $U$ is a field. Is it true that the quotient field of $\mathcal{O}_{\mathbb{C}^n,z}$ for $z \in U$ is a stalk of $\mathcal{K}$?

Let $U : = B_\epsilon (0) \subset \mathbb{C}^n$ and consider the ring $\mathcal{O}(U)$ of holomorphic functions on $U$. This is naturally contained in $\mathcal{O}_{\mathbb{C}^n,0}$. Is it true that the localization of $\mathcal{O}(U)$ at the prime ideal of all functions vanishing at the origin is $\mathcal{O}_{\mathbb{C}^n,0}$ ? Is it true that this prime ideal is not maximal?