Prove that for any field K, $\displaystyle t^3-3t+1$ 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: $\displaystyle \sum \alpha_i = \frac{-a_{n-1}}{a_n}$ where $\displaystyle \alpha_i$ are the roots, and $\displaystyle a_i$ the coefficients of the i-th power term in the polynomial, we get $\displaystyle \alpha_1+\alpha_2+\alpha_3=0$

Also using the formula for the product of the roots: $\displaystyle \prod \alpha_i=(-1)^n (\frac{a_0}{a_n}) \$ we get $\displaystyle \alpha_1\alpha_2\alpha_3=-1$

I only seem to be able to combine these equations, and the equations resulting from the fact that each $\displaystyle \alpha_i$ 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 $\displaystyle \alpha_1+\alpha_2+\alpha_3=0$) involving the three roots to combine with the previous to get a rational expression involving two roots.

Ideas?