Are there established algorithms for working with towers of low-degree algebraic extensions?

Question

I'm interested in doing 'computational ruler-and-compass' construction simulations along the lines of Euclidea and similar tools. Because the constructions can get rather involved, I'd like to be able to work with exact objects — for instance, in the construction of the 17-gon, this would include the value $\cos(2\pi/17)=\zeta_{17}+\zeta_{17}^{-1}$ where $\zeta_{17}=e^{2\pi i/17}$ is the pritive 17th root of unity; this value is constructed via quadratic extensions as part of the solution of the puzzle. Since I'm looking at ruler-and-compass style constructions, my extensions would be relatively low-degree — either degree-2, or maybe degree-3 if I consider doing some origami constructions. At the same time, the algebraic numbers themselves can get complicated — witness the aforementioned $\zeta_{17}+\zeta_{17}^{-1}$, which is of degree 8 and is usually built via a tower of three quadratic extensions from $\mathbb{Q}$. Are there any good algorithms for working with algebraic numbers that are in such (smallish) towers of low-degree extensions, particularly for constructing the towers and being able to assert algebraic equality etc.?