Recently I have been fascinated by properties of fields in ZF-models in which the Axiom of Choice (AC) does not hold.
For instance, in some such models it is consistent to say that the field of complex numbers has only two automorphisms!
My question: let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$. What can be said about $\mathrm{Aut}(\overline{\mathbb{Q}})$ in ZF-models without AC?
(I know one has to be very careful here, because algebraic closures of the rationals are not necessarily uniquely defined anymore in some models, if I am not mistaken.)