Choice is needed. For example, it is consistent with ZF that there exist an algebraic closure L of Q such that every absolute value on L is trivial (and Gal(L/Q) is trivial).
See: Hodges, Wilfrid. Läuchli's algebraic closure of $Q$. Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 2, 289--297. MR0422022