Algebraic complex numbers are countable without using fundamental theorem of algebra
I want to prove that the set of all algebraic numbers is countable. I know the proof where we show that the set of all polynomials with integer coefficients is countable (it is the union of $n$ tuples over countable $n \in \mathbb{N}$ and $\mathbb{Z}^n$ is countable), and then that the number of roots of each polynomial is finite, so that the algebraic complex numbers are a countable union of finite sets. I am wondering about other ways to do this.
Can I write any complex number $z=a+bi$ as $z = (1 + (a-1) + bi)$, and then for any polynomial that where $a_0z^n + a_1z^{n-1}+...+a_n = 0$ we have that $n + a_0 + a_1 + \cdots + a_n + N = 0$ for some real $N$ (but it will have to be an integer). And this is satisfied by a finite number of $N$ and $a_0,...,a_n$.
$\endgroup$ 4 Reset to default
