I want to factor $Y^5-5Y+12$ in $\mathbb Q[X]/(X^5-5X+12)$. Obviously $Y-X$ is a factor. Is there any good way to find the others? Or just a quick way to find all roots would be great aswell (in general).
So I want to factor $Y^4 + X*Y^3 + X^2*Y^2 + X^3*Y + X^4 - 5$ (into quadratics, if possible (should be possible)) (it doesn't have linear factors, as the polynomial with which we've build the quotient had only one real root to start with).
edit: Sorry, I falsely had a different polynomial here at start.
