Need example from number theory

[JOSEPH SHIPMAN]

I don’t want a necessary use of AC. I want an unnecessary use, that isn’t obviously unnecessary!

— JS

Sent from my iPhone

> On Aug 28, 2021, at 4:40 PM, Gyorgy Sereny <sereny at math.bme.hu> wrote:
> 
> 
> The most elementary example of the necessary use of AC I know
> (of course, it is not number theory):
> 
> (5.3) The union of a countable family of countable sets is countable.
> (By the way, this often used fact cannot be proved in ZF alone.)
> 
> To prove (5.3) let A_n be a countable set for each n \in N.
> For each n, let us _choose_ an enumeration (a_nk: k \in N) of A_n. [...]
> 
> (Thomas Jech: Set Theory
> Second Corrected Edition, p.39)
> 
> 
> Gyorgy Sereny
> 
> 
>> On Sat, 28 Aug 2021, JOSEPH SHIPMAN 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
>>