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 $\Q \sqbrk x$ 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:

$\Bbb A = \ds \bigcup_{\map p x \mathop \in \Q \sqbrk x} \set {x \in \C: \map p x = 0}$

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.

$\blacksquare$

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability
• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 2$: Equivalence of Sets. The Power of a Set: Problem $4$