Galois-theoretic derivation of the cubic formula

We are trying to find the roots r1,r2,r3 of the polynomialPlanetmathPlanetmath x3+ax2+bx+c=0. From the equation


we see that

a = -(r1+r2+r3)
b = r1r2+r1r3+r2r3
c = -r1r2r3

The goal is to explicitly construct a radical tower over the field k=(a,b,c) that contains the three roots r1,r2,r3.

Let L=(r1,r2,r3). By Galois theoryMathworldPlanetmath we know that Gal(L/(a,b,c))=S3. Let KL be the fixed field of A3S3. We have a tower of field extensions