Here is a countability argument that I like because it relies on almost nothing. Let be a ring.

**Theorem. **If is countable, then the polynomial ring is countable.

*Proof.* Since is countable, there is an injection . Let be prime numbers and consider the map

By unique factorisation in it follows that is an injection. Thus is countable.

We can use this to prove in a rather simple manner that

**Corollary 1. **The set of all algebraic numbers is countable.

*Proof.* It follows from the above that is countable. Let be a root of some minimal polynomial . We can assign to each a unique element as follows: if are the zeros of , assign to . This gives an injection from to , as desired.

More generally,

**Corollary 2.** A countable union of countable sets is countable.

*Proof.* Let be countable sets. Then there are injections for . Hence we have an injection

,

showing that is countable.