Need example from number theory

[Gyorgy Sereny]

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
>
>