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## Homework Statement

Let n a positive number, and let A

_{n}be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree

*n*. Using the fact that every polynomial has a finite number of roots, show that A

_{n}is countable.

## Homework Equations

Hint: For each positive number m, consider polynomials

[itex]\sum[/itex] a

_{i}x

^{i}from i=0 to n that satisfy

[itex]\sum[/itex] la

_{i}l ≤ m.

## The Attempt at a Solution

I want to prove this using induction, but I don't fully understand how to use the hint. How would I partition with algebraic numbers using it?

Starting using A

_{1}, which are the roots of polynomials, I know that these algebraic numbers are countable because

a

_{1}x

^{1}+a

_{0}=0 are the rational numbers, which are countable. What does the hint have to do with this?