It is implemented with polynomial rings!

I use SageMath to find minimal polynomials and solve the constructions algebraically. The reason I say that the library is only "arbitrary precision" and not exact/symbolic is that I use approximation in order to distinguish between points. I believe the "correct" way to implement this as exact would have been to save for each number its minimal polynomial and some interval that contains it and distinguishes it from the other roots of its minimal polynomial. It shouldn't be too much work to incorporate this, maybe I will do it some day :)