Set Theory – Proving Countability of Algebraic Numbers

Question

I am trying to prove that algebraic numbers are countably infinite, and I have a hint to use: after fixing the degree of the polynomial, consider summing the absolute values of its integer coefficients, and setting the sum less than or equal to $m$, for each $m \in \mathbb{N}$. I am also allowed to use the fact that every polynomial has a finite number of roots. I am not sure where to begin… Perhaps after fixing the degree of the polynomials, I could try enumerating them, and after proving countability of the set that contains the roots of all polynomials of a certain degree, I could state that the union of infinitely many ($\bigcup_{n=1}^{\infty}$, $n \in \mathbb{N}$) countably infinite sets is countable, and thus the algebraic numbers are countably infinite, too, but I am not sure if this is a good approach/how to enumerate the polynomials. Thanks!

Answer 1

The hint pretty much tells you what to do.

First, consider the polynomials of degree $1$ in which the sum of the absolute value of the coefficients is $1$; there's only finitely many of them, each with finitely many roots. (Think of this collection as corresponding to the pair $(1,1)$, the first $1$ giving the degree, the second $1$ giving the sum of the absolute value of the coefficients).

Then, consider the polynomials of degree $2$ whose sum of the absolute value of the coefficients is $1$; again, only finitely many, each with finitely many roots; make this set correspond to $(2,1)$.

Then, the polynomials of degree $1$ whose sum of the absolute value of the coefficients is $2$; only finitely many, each with finitely many roots. The set corresponds to the pair $(1,2)$.

Then consider the pair $(3,1)$; then the pair $(2,2)$; then the pair $(1,3)$. Then move on to the pair $(4,1)$, followed by $(3,2)$, followed by $(2,3)$, followed by $(1,4)$. Etc.

This will ensure that you cover each and every polynomial of positive degree with integer coefficients after finitely many steps; many algebraic numbers will occur more than once, but that's not a problem to showing they are at most countable (that they are infinite should be trivial).