Originally Posted by

**Opalg** The solution in the complex case will be similar to the previous problem. Let $\displaystyle \omega = e^{2\pi i/5}$, and for k=0,1,2,3,4 let $\displaystyle \lambda_k = u+\omega^{-k}v + \omega^{-2k}x + \omega^{-3k}y + \omega^{-4k}z.$

Then $\displaystyle (a + \omega^{k}b + \omega^{2k}c + \omega^{3k}d + \omega^{4k}e)^3 = \lambda_k$, and it follows that

$\displaystyle \displaystyle a = \tfrac15\sum_{k=0}^5\lambda_k^{1/3},\quad b = \tfrac15\sum_{k=0}^5\omega^k\lambda_k^{1/3},\quad c = \tfrac15\sum_{k=0}^5\omega^{2k}\lambda_k^{1/3},\quad d = \tfrac15\sum_{k=0}^5\omega^{3k}\lambda_k^{1/3},\quad e = \tfrac15\sum_{k=0}^5\omega^{4k}\lambda_k^{1/3}.$

Each $\displaystyle \lambda_k$ has three complex cube roots, and there are five of them, giving a total of $\displaystyle 3^5 = 243$ complex solutions. The number of real solutions is much more problematic. It's not clear to me whether there will be any at all.