Algebraic Numbers are Countable

Theorem
The set $\Bbb A$ of algebraic numbers is countable.

Proof
By definition, $\Bbb A$ is the subset of the complex numbers which consists of roots of polynomials with coefficients in $\Q$.

We can prove the theorem by a cardinality argument, counting the number of such polynomials and roots.

First note that the set $\Q \left[{x}\right]$ of polynomials over $\Q$ is countable.

Next, note that $\Bbb A$ can be written as the union of the set of roots of each polynomial, that is:
 * $\Bbb A = \displaystyle \bigcup_{p \left({x}\right) \mathop \in \Q \left[{x}\right]} \left\{{x \in \C : p \left({x}\right) = 0}\right\}$

Since each polynomial has a finite number of roots, $\Bbb A$ is a union of countably many finite sets.

From Countable Union of Countable Sets is Countable it follows that $\Bbb A$ is countable.