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
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.
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.