Let $\mathbb{Q}$ be the field of rational numbers, and let $\overline{\mathbb{Q}}$ be its algebraic closure. Assume $\overline{\mathbb{Z}}$ is the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}$, my question is that, what can we say about the integral domain $\overline{\mathbb{Z}}$?

Explicitly:

(1) Is it noetherian? (Hardly I think)

(2) How do we describe the nonzero prime ideals of $\overline{\mathbb{Z}}$ ? Are they maximal ? What are the residue fields? (I guess the residues are algebraic closure of finite field)

(3) Is it factorial?

(4) What is $\overline{\mathbb{Z}}^{\times}$ as an abstract group?

For (4), Obviously it contains all roots of unity, and as an abstract group it is isomorphic to $\mathbb{Q}/\mathbb{Z}$. Quotient it out, it becomes uniquely divisible, and hence an $\mathbb{Q}$-vector space. Can we find an explicit basis of it?

All these questions are clear for the integer ring $\mathcal{O}_K$ of a number field $K$, from algebraic number theory. And it is obvious that $\overline{\mathbb{Z}}$ is the direct limit of the integer ring of number fields, i.e $\overline{\mathbb{Z}}=\varinjlim_{K}\mathcal{O}_K$. But I am not quite familiar with the property of direct limit of rings. Can anyone give me some detail?

Additionally, one can ask the following question:

Consider $\overline{\mathbb{Q}}^{\times}$. It can be split into (not canonically) a product of a torsion part $\overline{\mathbb{Q}}^{\times}_{tor}$ and a free part $\overline{\mathbb{Q}}^{\times}/\overline{\mathbb{Q}}^{\times}_{tor}$ . The torsion part are exactly roots of unity, and is $\mathbb{Q}/\mathbb{Z}$. The quotient part becomes a $\mathbb{Q}$ vector space, too. What can we say about it?

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