Set of Algebraic Numbers is Countable

This result is particularly useful in providing a proof for the existence of transcendental numbers, without the need to construct any particular example. For the proof we will use the fact that a countable union of countable sets is countable, and thus we first desire the following lemma.

Proof

Write

Each set in this union is countable, because we can construct a natural bijection between the set of polynomials of degree $n$ and ${\mathbb{Q}}^{n+1}$ by

and ${\mathbb{Q}}^{n+1}$ is countable because a finite cartesian product of countable sets is countable, and $\mathbb{Q}$ is countable. Therefore $\mathbb{Q}[X]$ is written as a countable union of countable sets, and is therefore countable.

We can now prove the main theorem.

Proof

We may write the set of algebraic integers as

and by the fundamental theorem of algebra, since $\mathbb{Q}$ is a subfield of $\mathbb{R}$, we have that

Thus we have written the set of algebraic integers as the countable union of countable sets, which is countable.