Fields of characteristic $p$ from rings of integers

Question

I have some questions about fields of characteristic $p$ arising from rings of algebraic integers.

If $K$ is a number field and $\mathcal{O}_K$ is its ring of integers, and $M$ is a maximal ideal in this ring, then we know that $\mathcal{O}_K/M$ is a finite field (of characteristic $p$).

(1) Suppose $\mathcal{O}$ is the full ring of algebraci integers in $\mathbb{C}$ and $M$ a maximal ideal containing a prime (rational) integer $p$. Then $\mathcal{O}/M$ is a field of characteristic $p$. How do we ensure that it is infinite?

(2) If $M'$ is another maximal ideal containing $p$, then are $\mathcal{O}/M$ and $\mathcal{O}/M'$ isomorphic?

(I saw introduction of these fields in some book on modular representation theory of groups, but I didn't find these question at least to be stated as exercises)

Answer 1

For $(1)$ you can easily show that $\mathcal O/M$ is algebraically closed, hence it is infinite.

It is also algebraic over $\mathbb F_p$, hence it is the algebraic closure. This answers $(2)$.