Prove that for any field K, splits or is irreducible in K.
Hint: show that any zero is a rational expression in any other zero.
Using the formula for the sum of roots: where are the roots, and the coefficients of the i-th power term in the polynomial, we get
Also using the formula for the product of the roots: we get
I only seem to be able to combine these equations, and the equations resulting from the fact that each is a root of our polynomial to get equations of two variables where a power higher than 1 is present in each variable, and quadratic formulas don't seem to resolve those nicely. I think if I'm reading the problem right, we don't want any roots in our relating equation so I guess we need another rational expression (other than ) involving the three roots to combine with the previous to get a rational expression involving two roots.