So, the field of complex algebraic numbers is algebraically closed, meaning that every root of a polynomial with algebraic coefficients is also algebraic.
But algebraic numbers are defined as roots of polynomials with integer coefficients.
So does that mean that every polynomial with algebraic coefficients can be transformed into a polynomial with integer coefficients that has the same roots? Or is it a system of integer polynomials with each corresponding to a different root of the initial polynomial?
I changed the title of the question a little so it makes more sense.