# Chevalley's valuation extension theorem and the axiom of choice

Hello,

Do we know if the axiom of choice is needed for Chevalley's valuation/place extension theorem (i.e. the theorem that states that for every valued field and a field extension, one can extend the valuation to the field extension)?

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Can somebody more expert about this retag? I'd say commutative algebra and/or algebraic number theory, but I don't really know. –  darij grinberg Mar 27 '11 at 18:08

I think this more or less answers it?

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What about finite extensions? (Disclaimer: I don't know any proof of the theorem, even for finite extensions.) –  darij grinberg Mar 27 '11 at 18:07
I would like to see a bit more in an answer than is provided by the answers to the linked question. They say that the answer is "yes": one needs AC (that is my guess as well), but unless I missed it no argument or reference is given. –  Pete L. Clark Mar 28 '11 at 6:00
@darij: for a finite extension, there are only finitely many ways to extend the valuation, so surely AC is not needed. –  Pete L. Clark Mar 28 '11 at 6:00

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 thelemma' 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.

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