Need example from number theory

Buzzard, Kevin M k.buzzard at
Sun Aug 29 08:05:27 EDT 2021

Two non-answers:

  1.  the proof of Fermat's Last Theorem uses AC in the intermediate functional analysis needed to prove the Langlands-Tunnell theorem, or at least AC is present everywhere in expositions of this, basically because the generic number theorist interested in the analysis involved in the Langlands Programme has no interest at all in removing AC from it. However this is not suitable for undergraduates.

  1.  Proofs in UG level topology sometimes use AC in unnecessary (but not obviously unnecessary) ways, for example in the proof that a compact subspace of a Hausdorff space is closed in my UG lecture notes it says "for a point x not in the subspace, then for each y in the subspace one can choose disjoint opens containing x and y. Now let y vary and take a finite subcover". People not willing to accept AC as part of mathematics can of course now argue over what the exact definition of Hausdorff should be, because whether I just used AC or not depends on the small print. This is UG level but not number theory.

This sort of question is difficult in general because basic concepts like "closure of a subspace of a metric space" has two definitions which only coincide under something like countable dependent choice, so with no AC at all in fact one can argue that a lot of what is being told to undergraduates is imprecise in the sense that in the absence of AC one has to be more careful about which of the now-no-longer-equivalent definitions one uses.

From: FOM <fom-bounces at> on behalf of JOSEPH SHIPMAN <joeshipman at>
Sent: 28 August 2021 16:40
To: Foundations of Mathematics <fom at>
Subject: Need example from number theory

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What is an example of a theorem of number theory which has a well-known proof *suitable for undergraduates*, that uses the Axiom of Choice in a way that is not obviously unnecessary?

We know that AC can be eliminated from the proof of any arithmetical statement, but I’d like an example I can easily explain.

— JS
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