Solution

Solve the quadratic resolvent:

u^{3}, v^{3} = \frac{- 27 b \pm - 27 Disc ( f )}{2} .

(100)

Choose cube roots with

uv = - 3 a

. Inverting the discrete Fourier transform gives

α_{1} = \frac{u + v}{3}, α_{2} = \frac{ω ^{2} u + ω v}{3}, α_{3} = \frac{ω u + ω ^{2} v}{3} .

(101)

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.

Solved by gpt-5.6-sol high.

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