Let's consider a set of numbers that one knows to high precision, and one knows or has a strong suspicion that `exact versions of these numbers' (see below) belong to a real extension of $\mathbb{Q}$. Is there an algorithm that allows one to find the rational coefficients? Even having a slow algorithm that is better than a brute force search would be interesting. Or is it known that this is not possible in general?

To be concrete, let us consider the extension $$ E = \mathbb{Q} (\cos(\pi/21), \cos(5\pi/21), \cos(11\pi/21), \cos(13\pi/21), \cos(17\pi/21), \cos(19\pi/21)) \ , $$ (note that $$\cos(\pi/21)+\cos(5\pi/21)+\cos(11\pi/21)+\cos(13\pi/21)+\cos(17\pi/21)+\cos(19\pi/21) = -\frac{1}{2} \ ,$$ so only six coefficients are required).

I know that the set of numbers I obtained numerically can be written in terms of the extension above. They are in fact the squares of $F$-symbols, satisfying the pentagon equations. By a brute-force search for coefficients with small denominators, one can identify some of the numbers. The ones found can then be used to find the others, by making use of the pentagon equations. However, for many of the numbers, both the numerators and denominators of the coefficients become large (up to 10^13 in the given example, so one needs the numbers to high precision to start with), and the procedure is very cumbersome. In more complicated situations, my brute force approach is not powerful enough, which is why I'm asking if a more sophisticated method/algorithm exists.

So more precisely, the question would be if there is an algorithm that finds an element $e \in E$, where $E$ is a real extension of $\mathbb{Q}$, such that $| e - r | < \epsilon$, for a given $r \in \mathbb{R}$ and precision $\epsilon$.

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