The set of algebraic numbers is countable.

I'm currently working through Rudin's PMA, and I'm on exercise 2.2 which tasks me to prove that the set of algebraic numbers is countable. There is a hint we are given to use, which is that the number of equations that satisfy the equation$n + |a_0| + \cdots + |a_n| = N$ for any positive integer $N$ is finite. Now, here was my thought process:

1. Establish a set of $(n + 1)$-tuples consisting of the elements $(n, a_0, a_1, \ldots, a_n)$, where $n$ is a positive integer, and $a_i$ is any integer, not all $0$, such that the hint is satisfied; call these sets $C_N$.
2. Let $P_N$ be the set of polynomials with complex inputs whose coefficients satisfy the above hint. Then $P_N$ is finite.
3. Let $R_N \subset P_N$ be the set of polynomials that have a root at $z = 0$. Since $P_N$ is finite, then $R_N$ is finite.
4. Let $A_N$ denote the set of complex numbers that satisfy a polynomial in $R_N$. Since any $n$-degree polynomial has at most $n$ roots, then $A_N$ is also finite.
5. Take the union of $A_N$ over all positive integers $N$. Then this set is countable, since a union of countable sets is countable.

Please let me know if my logic is right!