Need example from number theory

Raphael Douady rdouady at
Sun Aug 29 00:49:45 EDT 2021

Gyorgy's example is a good case of *unnecessary* use of AC.
The school case where it is necessary is Tychonoff theorem and Czech
compactification: any product of compact sets for the product topology is
compact. Unlimited consequences in functional analysis.
In number theory, it implies the existence of an algebraic closure for any
field, and also the incomplete base theorem, which is very important in
number field theory.

On Sun, Aug 29, 2021 at 12:42 AM Raphael Douady <rdouady at> wrote:

> Joe,
> My father's book on Galois theory starts with Zermelo and choice axioms.
> Then goes on with Galois theory. I'm sure you'll find what you need there.
> Raphael
> On Sat, Aug 28, 2021 at 2:50 PM JOSEPH SHIPMAN <joeshipman at> wrote:
>> 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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