Dirichlet's unit theorem tells us that for any number field $K$, it is well known that the unit group $\mathcal{O}_{K}^{\times}$ of the ring of integers $\mathcal{O}_{K}$ of $K$ is isomorphic to $\mu(K)\times \mathbb{Z}^{r+s-1}$, where $\mu(K)$ is a (finite) group of root of unities and free part corresponds to fundamental units.
My question is: is there any generalization of the theorem for infinite algebraic extensions? For example, let $\overline{\mathbb{Z}} = \mathcal{O}_{\overline{\mathbb{Q}}}$ be a ring of algebraic integers. Can we describe $\overline{\mathbb{Z}}^{\times}$? I think it should be $(\mathbb{Q}/\mathbb{Z})\oplus\left(\bigoplus \mathbb{Q}\right)$, where the direct sum has countably many copies of $\mathbb{Q}$, because it may be a direct limit of $\mathcal{O}_{K}^{\times}$. Is this true? If it is true, can we write element of $\overline{\mathbb{Z}}^{\times}$ that corresponds to each $\mathbb{Q}$-summands? Thanks in advance.