Proving algebraic numbers are countable?

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The discussion centers on proving that the set of algebraic numbers, specifically those derived from polynomials with integer coefficients of degree n, is countable. It emphasizes that each polynomial has a finite number of roots, and to establish countability, one must show that there are only countably many such polynomials. The hint suggests using a condition on the coefficients to count the polynomials effectively, leading to a finite number of roots for each polynomial. By taking the union of these finite sets over all possible values of m, the overall set of algebraic numbers can be shown to be countable. The conclusion reinforces that the union of countable sets remains countable, solidifying the proof.
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Thank you for your help (and patience), I know it took a while for me to get it.
 

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