Obviously, algebraic numbers uniquely determine their minimal polynomials but not the other way around. But, in general, what is the worst case scenario- if given a minimal (irreducible, monic, of least degree) polynomial, how many algebraic numbers can be its zeroes? Is there any way to consistently (if arbitrarily) single out/pick out a special zero of these from which the other zeroes may be found by a single given and universal operation (an operation that would work for/generatr any such set of zeroes, if given this special one among them)? For example, simply selecting the first zero nearest to the positive real axis (and breaking ties by choosing the one on the counterclockwise side thereof) might work to single out (arbitrarily) a special zero, and then the others may be specified by some fixed rotation around 0 determined uniquely by the special zero. Would this method work? Are there others?
Note: basically, I am trying to think of a way to specifically name a single algebraic number uniquely by telling its minimal polynomial over the field of algebraic numbers.