Real solving univariate polynomials in an extension field

Recently I am interested in studying the algorithmic and implementation details of the problem of isolating the real (and if possible the complex) roots of univariate polynomials with coefficients in an a single extension field. Moreover, I am considering exact algorithms only, i.e. algorithms that allow only operations with integers of arbitrary bit size.

I am considering the easy case, where the coefficients of the polynomial, say , are depending on one real algebraic number, say , i.e. , where is the variable. The number is in isolating interval representation, i.e. , hence in order to represent it, we use a square-free polynomial, , and an isolating interval that contains and no other real root of . Let is the degree of and be the maximum bit size of the coefficients of .

It seems that there are two approaches to our problem:

1. Compute a polynomial with integer coefficients that has as roots the roots of (and probably other roots as well) and solve it.
2. Apply to the well known algorithms for univariate real root isolation, e.g. Sturm, Descarte, Bernstein, CF, to , suitably modified so that to work with polynomials with real algebraic numbers as coefficients.

What is the best way from a theoretical and an implementation point of view, in order to isolate the real roots of $f$? Is there any other approach different from the ones mentioned above?