## Axiom of Choice and Number Theory [closed]

There are so many applications of the Axiom of Choice (and consequently its equivalents) in number theory. But do you know any application of the Zorn's Lemma in Number Theory !? I mean a theorem or a problem in this field that uses Zorn's Lemma ?

Edit: of course any application of AC is equivalent to an application of the Zorn's Lemma. But I am sure you agree that some proofs work better with AC, some work better with the Well ordering axiom and some work better with Zorn's Lemma. When we prove that any vector space has a base we use Zorn's lemma. Nobody writes a proof with well ordering axiom while anybody can write such a proof.

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Why do people vote up questions like this? – S. Carnahan Jan 15 at 10:53
Jekteem, please give us an example of an application of the Axiom of Choice in number theory, and explain why it is not equivalently an application of Zorn's lemma. – S. Carnahan Jan 15 at 10:54
Hindman's theorem has a proof using Zorn's lemma. See, e.e., here: math.toronto.edu/lgoldmak/Hindman.pdf – Péter Komjáth Jan 15 at 12:41
And I suppose existence of maximal ideals does not count as "number theory"? – Andrej Bauer Jan 15 at 12:56
Concerning the last sentence of the question and Emil's comment: I sometimes find it helpful to use transfinite induction instead of Zorn's Lemma because it makes it easier to think (and write) about interleaving the construction with new sorts of steps, in order to obtain some additional properties of the object being constructed. – Andreas Blass Jan 15 at 13:52
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