[FOM] Algebraic closure of Q

[Timothy Y. Chow]

The following question was motivated by a recent discussion on 
sci.math.research.

Hodges showed that ZF doesn't prove the uniqueness (up to isomorphism) of 
an algebraic closure of Q (W. Hodges, "Lauchli's algebraic closure of Q," 
Math. Proc. Camb. Phil. Soc. 79 (1976), 289-297).

Question: What's really needed to prove the existence and uniqueness of 
Qbar?  (There might be more than one answer to this question.)

Tim