Are The Algebraic Numbers Countable?

$\mathbb{Q}$ , rational numbers set is a countable infinite set. I mean, there exists a bijective function $f:\mathbb{N} \rightarrow \mathbb{Q}$ where $\mathbb{N}$ is natural numbers set. Now, each algebraic number $\alpha$ is a root of a polinomyal $p(x)$ with rational coefficients. Let $n$ be the degree of the polynomial $p(x)$ . Then every polynomial with rational coefficients of degree $n$ can be seen as a element of $\mathbb{Q}^{n + 1}$ where $|\mathbb{Q}^{n + 1}|$ is countable infinite due to a finite product of countable sets is a countable set. By this way, all polynomials with rational coefficients can be seen as contained in $\cup_{i =1}^\infty \mathbb{Q}^{i}$ , with $i \in \mathbb{N}$ and every set of this union is a countable infinite set. Therefore, due to countable infinite union of countable infinite sets is a countable infinite set, $\cup_{i=1}^\infty \mathbb{Q}^i$ is a countable infinite set, and then we can say that the set of the all polynomial with rational coefficients is a countable infinite set,and because of this we can infire that the set of algebraic numbers $S$ is a countable infinite set, because $\text{cardinal of Q} = |\mathbb{Q}| \leq \text{cardinal of S} = |S| \leq$ $\leq|\{ \text{polynomial with rational coefficients}\}| \leq |\cup_{i=1}^\infty \mathbb{Q}^i| = |\mathbb{Q}|$ .

(Applying Schroeder-Bernstein Theorem.- If A and B are sets and $|A| \leq |B| \leq |A|$ , then |A| = |B|.)

For each degree starting from one , we can make a list of all the possible coefficients set since it is countably infinite i.e. at first we create a list of all the coefficients set of degree 1 polynomial and then degree 2 polynomial and so on. Hence,the number of rational polynomials are countably infinite.

Now for each of the algebraic numbers there exist some coefficients set (more than one ).But we only define a mapping from algebraic numbers to the minimal polynomial which has the number as the root.In this way,there is a bijection between all those countably infinite polynomials and the algebraic numbers

Hence in this way we can make a bijection between natural numbers and algebraic numbers ,the answer is indeed countably infinite.

My explanation is motivated by this wiki cardinality Specifically the part where they show the bijection between the rational and natural numbers.