Algebraic Numbers are Countable

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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.

By Set of Polynomials over Infinite Set has Same Cardinality, 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:

$\Bbb A = \displaystyle \bigcup_{p \left({x}\right) \mathop \in \Q \left[{x}\right]} \left\{{x \in \C : p \left({x}\right) = 0}\right\}$

From Polynomial over Field has Finitely Many 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.

$\blacksquare$


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