Ok so I have this as the prompt. Fixing $n \in \Bbb{N}$ let $A_{n}$ be the set of algebraic numbers obtained as roots of polynomials with integer coefficients that have degree $n$. Using the Fundamental Theorem of Algebra we know that every polynomial has a finite number of roots, show that $A_{n}$ is countable.
Hint: For each $m \in \Bbb{N}$, consider polynomials $a_{n}x^{n}+a_{n-1}x^{n-1}+ \cdots +a_{1}x+a_{0}$ that satisfy $|{a_n}| + |{a_{n-1}}| + \cdots + |{a_1}| + |{a_0}| \leq m$
I am lost as to where to start, and the hint I guess is lost on me.
Could we use the fact that since $|{a_n}| + |{a_{n-1}}| + \cdots + |{a_1}| + |{a_0}| \leq m$ all are in $\Bbb{R}$ we can say for each $m\in \Bbb{N} $ ,$-m \leq {a_n} + {a_{n-1}} + \cdots + {a_1} + {a_0} \leq m$
I suspect that using the fundamental theorem of algebra and finite roots is a clue towards establishing that this implies $An$ is denumerable and from that countable but not clear on proper. Sorry if this is all over the place I have been spinning my wheels for days on this.