I'm trying to determine the minimal polynomials of $-\alpha$, $1-\alpha$, $2\alpha$, and $1/\alpha$, given that the minimal polynomial of $\alpha$ is $x^3-x-1$.
Since $x^3-x-1$ is the minimal polynomial of $\alpha$, we have $\alpha^3-\alpha-1=0$. So, $\alpha^3=\alpha+1$, and $-(\alpha^3-\alpha-1)=0$, so $-\alpha^3+\alpha+1=0$, and thus $-\alpha$ satisfies $x^3+x+1=0$. Would that be the minimal polynomial? How would I get the other minimal polynomials requested? Thanks!