Let n a positive number, and let An 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 An is countable.
Hint: For each positive number m, consider polynomials
[itex]\sum[/itex] aixi from i=0 to n that satisfy
[itex]\sum[/itex] lail ≤ 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 A1, which are the roots of polynomials, I know that these algebraic numbers are countable because
a1x1+a0=0 are the rational numbers, which are countable. What does the hint have to do with this?