Algebraic numbers and their minimal polynomials

Question

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.

Answer 1

An irreducible polynomial $f$ of degree $n$ (over ${\mathbb Z}$, say) has $n$ distinct roots (in $\overline{\mathbb Q} \subseteq {\mathbb C}$). You could single out a specific root by giving a (rational) approximation, say an element of $q \in {\mathbb Q}[i]$ and a radius $r \in {\mathbb Q}, r > 0$.

So an algebraic number $z$ can be represented by a triple $(f,q,r)$. (To be clear: only $f$ is uniquely determined by $z$, but $q$ and $r$ are not; and not all triples are valid representations of an algebraic number).

This representation can be used to do exact computations with algebraic numbers (for instance by computer algebra systems). For example, to add two algebraic numbers $z_1$ and $z_2$ with representations $(f_1, q_1, r_1)$ and $(f_2, q_2, r_2)$, you first compute the minimum polynomial $f$ of $z_1 + z_2$ (there are standard algorithms to do this) and take $(f, q_1 + q_2, r_1 + r_2)$ as representation of the sum. (You may be unlucky and there may be another root of $f$ within distance $r_1 + r_2$ of $q_1 + q_2$; in that case you first have to get tighter rational approximations of $z_1$ and $z_2$).

Answer 2

I'm kind of confused as to what you're asking. A polynomial of degree $n$ has at most $n$ roots. They're all algebraic by definition. Might be you have repeated roots even when the polynomial is irreducible, but that will only happen in special circumstances, of which you might not be aware.

Depending on the polynomial, these roots can all be 'identical' in a sense, or very different. What you're suggesting seems to hint at the Galois group of that polynomial, where a cyclic Galois group (as in the case of cyclotomic polynomials $\Phi_n$, e.g.) means all the roots are 'the same', and a Galois group more like $S_n$ means they are all 'different'. Does this help?