I'm having difficulties understanding my textbook's decription of solving cubic equations using Lagrange Resolvents and symmetric polynomials.
Here's what I understand:
We can also write:
where are the cubic roots of 1. I then understand that and are symmetric polynomials in r, s, and t. It is also solvable that the elemntary symmetric functions in r, s, t are 0,p,q where
The part where I get confused is that the book claims that and are the roots of the quadratic polynomial , which seems obvious to me. Then they claim you can use the quadratic formula to solve for and in terms of p and q, thus allowing you to solve a system of equations to acquire r,s,t.
How can you use the quadratic formula to "explicitly solve for and in terms of p and q"?