I am reading a cubic formula derivation here:
http://danielrui.com/papers/cubicPolynomial.pdf
It looks fairly straight forward. The author defined:
$y = \sqrt[3]{u} − \sqrt[3]v$
so far so good, I suppose there can always be a $u$ and $v$ that will work. But then, he proceeds with
"Let's define $v − u = e$ ('cause we can)"
I am lost here. $e$ is not just any constant. It is
$e = -\frac{a^3}{27}+\frac{a^3}{9}-\frac{ab}{3}+c$
How can he make that claim?
When you make $y=\sqrt[3]{u}-\sqrt[3]{v}$ you are simply renaming your variables. Nothing prevents you from assuming that this two new variables also satisfy another relation between them (think of this as if you go from one single equation to a system of equations). In this particular case, the most clever relation to impose between $u$ and $v$ is $v-u=e$. Note that $e$ does not depend on $y$, as a comment points out.