Solve $x^3 - x + 1 = 0$ Solve $x^3 - x + 1 = 0$, this cannot be done through elementary methods.
Although, this is way out of my capabilities, I would love to see a solution (closed form only). 
Thanks!

Let $$x=\frac{a}{y}+y$$ then $$y=\frac12 \left (x+\sqrt{x^2-4a} \right )$$
Then you have $$1-y-\frac{a}{y}+\left (y+\frac{a}{y}\right )^3 =0\stackrel{\cdot y^3}{\iff} y^6+(3a-1)y^4+y^3+(3a^2-a)y^2+a^3=0\;\stackrel{a=\frac13,z=y^3}{\iff}\; z^2+z+\frac{1}{27}=0$$
Solve and rewind all the substitutions(not the $a$ one, I kept it along to see what would eventually fit). Didn't pay attention to any restrictions so you'll have to check each value too. I would like to know where my method is non-elementary too.
PS: The $a$ part is to show why this Vieta's substitution variant works.
