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The following facts seem to be part of the folklore (where $\mathsf{ZF}$ means Zermelo-Fraenkel set theory with no axiom of choice):

  1. it is consistent with $\mathsf{ZF}$ that there exists an uncountable algebraic closure of $\mathbb{Q}$ and of $\mathbb{F}_p$,

  2. $\mathsf{ZF}$ proves that every countable algebraic closure of $\mathbb{Q}$ or of $\mathbb{F}_p$ is isomorphic (to the “standard” one, if you will).

Neither is too difficult to prove: statement (1) can be obtained in $\mathsf{ZFA}$ fairly easily (take a copy of the “standard” algebraic closure as set of atoms, with the full absolute Galois group as group of permutation acting on these atoms, and use the filter of subgroups generated by the stabilizer of finite extensions of the prime field to construct a permutation model of $\mathsf{ZFA}$) and is a transferable statement so the Jech-Sochor theorem applies; as for (2), the isomorphism can be constructed by a standard back-and-forth argument, using the countability to pick the smallest element at each step. If there is any subtlety I missed in these arguments, please let me know.

Regarding (1), the fact that it is consistent with $\mathsf{ZF}$ that there exists an algebraic closure of $\mathbb{Q}$ which is not the standard one is easy to find a reference for: this is originally proved in Läuchli, “Auswahlaxiom in der Algebra”, Comment. Math. Helv. 37 (1962/1963) 1–18, and a commentary with further explanations is given in Hodges, “Läuchli's algebraic closure of $\mathbb{Q}$”, Math. Proc. Cambridge Philos. Soc. 79 (1976) 289–297. Läuchli's paper (loc. cit., 2.1) does mention that the algebraic closure in the constructed model is uncountable, but Hodges makes no remark on the subject as far as I can see. Of course, granted statement (2) above, an algebraic closure of $\mathbb{Q}$ that is not the standard one is necessarily uncountable. Anyway, this leaves the corresponding question for $\mathbb{F}_p$ unanswered. Another possible partial reference is Asaf Karagila's preprint “Iterated Failures of Choice”, ¶5.2 (esp. the remark that “the symmetric copy obtained will have a distinct cardinality of any ground model algebraic closure”).

There are, of course, various online references to either of the two facts above, in particular here on MathOverflow or on MSE: see Andrés E. Caicedo's comment and Zhen Lin's answer on this MSE question, as well as this old Usenet thread and this MO question for various related matters.

Still, none of this is fully satisfactory. Is there a citable reference somewhere in the literature (preferably a textbook or survey paper) where the statements (1) and (2) above are written black on white, either exactly as I have stated them, or in a form that trivially implies them?

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    $\begingroup$ I mean, (2) is a very easy consequence of the ZFC result that all algebraic closures of a fixed field are isomorphic. If $F_1,F_2$ are two well-orderable algebraic closures of a well-orderable field $K$, there are sets of ordinals $A_1,A_2$ and $A_K$ which code these structures, then in $L[A_1,A_2,A_K]$ the field $K$ exists, and both $F_i$ exist and are algebraically closed, so they are isomorphic. Of course, if $K$ is $\Bbb Q$ or $\Bbb F_p$ we can ignore $A_K$ altogether, and if $F_i$ is countable, we can just take $A_i$ to be some subset of $\omega$. $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 2, 2023 at 11:27

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