I want to find a way to describe all the extensions of $\overline{\mathbb{Q}}^\times$ by $\overline{\mathbb{Q}}$, i.e., all the abelian groups $A$ (and the maps $\alpha$ and $\beta$) that fit into the exact sequence of abelian groups
$$0 \to \overline{\mathbb{Q}} \overset{\alpha}{\to} A \overset{\beta}{\to} \overline{\mathbb{Q}}^\times \to 0.$$
Of course $A:=\overline{\mathbb{Q}} \times \overline{\mathbb{Q}}^\times$ with the obvious maps works, but that's always the case. I was thinking about using the exponential map, but the problem is that the exponential of an algebraic number is in general not algebraic...
I could also compute $Ext^1(\overline{\mathbb{Q}}^\times, \overline{\mathbb{Q}})$ using resolutions, but I am not sure whether this is easier...
Both groups are divisible, hence injective. Since $\bar{\mathbb Q}$ is injective $Ext^1(-,\bar{\mathbb Q}) = 0$, and there are no other extensions. (There are actually no extensions the other way either.)