Is the Set of All Algebraic Numbers Countable?

[alphonsas]

Homework Statement

A complex number z is said to be algebraic if there are integers
a0; a1...; an not all zero such that z is a root of the polynomial,
Prove that the set of all algebraic numbers is countable.

Homework Equations

The Attempt at a Solution

For every natural number N there are only finitely many such polynomials

But how to prove the set is countable

 

[Office_Shredder]

For every natural number N there are only finitely many such polynomials

Finitely many such polynomials that do what?