Countability of Algebraic Numbers

Be $P_n$ the set of all polynomials with integer coefficients and degree less or equal to $n$. We can observe that exists a bijection between $P_n$ and $\mathbb{Z}^{n+1}$, as we can write any polynomial as $a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$ So we can express the polynomial as a tuple $(a_0, a_1, a_2, \ldots, a_n)$. As the product of countable sets is countable, we can conclude that $P_n$ is countable.

As each polynomial has a finite number of roots, the set:

$A_n = \{x \in \mathbb{R} : x \text{ is a root of some polynomial in } P_n\}$

Then, be $H_n = \bigcup_{i=0}^{n} A_i$, we can see that $H_n$ is countable, as it is the countable union of countable sets.

Finally, be $H = \bigcup_{n \in \mathbb{N^*}} H_n$, we can see that $H$ is countable, as it is the countable union of countable sets. And $H$ is the set of all algebraic numbers, so we have proved that the set of algebraic numbers is countable.