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, we can ask the following question:

Is it decidable (i.e. computable by a Turing machine) whether two real irrational algebraic numbers $\alpha$ and $\beta$ generate the same extension of the rationals?

A related question is a reachability question in dynamical systems:

If $\alpha, \beta$ are real irrational algebraic numbers, does there exist an effective procedure for determining whether there exists some integer $k \geq 1$ such that $$ k\alpha = \beta \mod 1? $$

For example, when $x = \sqrt{2}/4$, and $y = \sqrt{2}/2$, we can find that there exists a $k=2$ solving the problem. But it is unclear if this is the case in general.

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