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

As for the origin of "Zorn's Lemma", try

Campbell, Paul J. The origin of "Zorn's lemma''. Historia Math. 5 (1978), no. 1, 77--89. MR0462876

The following is quoted from Berrick and Keating, 2000, p26: The name of the statement [Zorn's Lemma], although widely used (allegedly first by Lefschetz), has attracted the attention of historians (Campbell 1978). As a `maximum principle', it was first brought to prominence, and used for algebraic purposes in Zorn 1935, apparently in ignorance of its previous usage in topology, most notably in Kuratowski 1922. Zorn attributed to Artin the realization that the `lemma' is in fact equivalent to the Axiom of Choice (see Jech 1973). Zorn's contribution was to observe that it is more suited to algebraic applications like ours.} is equivalent to the Axiom of Choice, and hence independent of the axioms of set theory.