It is well-known that arbitrary angles cannot be trisected by using a straightedge and a compass. The proof of this fact is done by reduction to unsolvability of cubic equations with square roots (see e.g. Trisecting angle equivalence of constructing a segment). For example, the angle of 40 degrees cannot be constructed, because it involves the solutions of $4x^3 - 3x - \cos(120^\circ) = 0$, which cannot be represented with square roots.
However, this reduction may be unreversible: even if we could trisect arbitrary angles, it is not obvious that we can construct solutions of arbitrary cubic equations with a straightedge and a compass (and I suspect it is false). So I am wondering the relative difficulty of the following three problems:
- trisection,
- solving arbitrary cubic equations = calculating cube roots of arbitrary complex numbers, and
- calculating cube roots of arbitrary real numbers.
(I use the word solve meaning constructing the solution of a specific equation, and calculate too. Coefficients of equations are given as coordinates of existing points.)
It can be proven that solving arbitrary cubic equations is as difficult as calculating cube roots of complex numbers (the equivalence in 2.). We can construct 2. if we can construct 1. and 3., and we can construct both 1. and 3. if we can construct 2. I am wondering if 1. and 3. are independent (given a straightedge and a compass).