Solution
= Solution
Solve the quadratic resolvent:
$$
u^3,v^3=\frac{-27b\pm\sqrt{-27\operatorname{Disc}(f)}}2.
$$
Choose cube roots with $uv=-3a$. Inverting the discrete Fourier transform gives
$$
\alpha_1=\frac{u+v}{3},\qquad
\alpha_2=\frac{\omega^2u+\omega v}{3},\qquad
\alpha_3=\frac{\omega u+\omega^2v}{3}.
$$
These expressions use only square and cube roots. A general cubic is first translated to remove its quadratic term, proving the existence of a formula by radicals.