# 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