Galois-theoretic derivation of the cubic formula

We are trying to find the roots $r_{1},r_{2},r_{3}$ of the polynomial $x^{3}+ax^{2}+bx+c=0$ . From the equation

(x-r_{1})(x-r_{2})(x-r_{3})=x^{3}+ax^{2}+bx+c

we see that

$\displaystyle a$	$\displaystyle=$	$\displaystyle-(r_{1}+r_{2}+r_{3})$
$\displaystyle b$	$\displaystyle=$	$\displaystyle r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}$
$\displaystyle c$	$\displaystyle=$	$\displaystyle-r_{1}r_{2}r_{3}$

The goal is to explicitly construct a radical tower over the field $k=\mathbb{C}(a,b,c)$ that contains the three roots $r_{1},r_{2},r_{3}$ .

Let $L=\mathbb{C}(r_{1},r_{2},r_{3})$ . By Galois theory we know that $\operatorname{Gal}(L/\mathbb{C}(a,b,c))=S_{3}$ . Let $K\subset L$ be the fixed field of $A_{3}\subset S_{3}$ . We have a tower of field extensions