For an algebraic number $\alpha$, let $f_\alpha$ denote its minimal polynomial. We can symbolically represent an algebraic number $\alpha$ by the tuple $$ (f_\alpha, x, y, r) \in \mathbb{Q}[x] \times \mathbb{Q}^3 $$ so that $\alpha$ is the unique root of $f_\alpha$ inside the circle centered at $(x, y)$ of radius $r$ in the complex plane -- that is $x, y, r \in \mathbb{Q}$ give the information needed to distinguish $\alpha$ from the other roots of $f_\alpha$ (Galois conjugates).
Given these finite representations of algebraic numbers, I am curious about the following question:
Can we effectively compute the square roots of an algebraic number? That is, given the finite representation of an algebraic number as above, can we find the square roots expressed in the same finite representation?
It is known that with this finite representation of algebraic numbers we can effectively check equality, and multiply, add, divide, etc., but it is not obvious to me if we can compute square roots. When the number is rational, we can compute its square root and obtain such a representation as above via the polynomial $x^2 - a = 0$, where $a \in \mathbb{Q}$, but outside of rationals it is not obvious to me, although I get the sense I am missing something basic here.
