I’m working on the following question.
Let $f(x) = x^3-x+1$ with root $\alpha$. Find the minimum polynomial for $\beta = 2 - 3\alpha + 2\alpha^2$.
My question is about how you might go about finding the minimum polynomial for $\beta$ and how, in general, one might find the minimum polynomial for equations of a primitive element.
Thanks in advance.
Since it is an extension of prime degree $3$ either $\beta$ is rational (and an integer since it is an algebraic integer) or its minimal polynomial has degree $3$.
We know it is the latter case because $\Bbb{Q}[\alpha]=\Bbb{Q}\oplus\alpha\Bbb{Q}\oplus\alpha^2\Bbb{Q}$.
Then the minmial polynomial is $\det(B-tI)$ where $B$ is the matrix of the multiplication by $\beta$ on $\Bbb{Q}\oplus\alpha\Bbb{Q}\oplus\alpha^2\Bbb{Q}\cong \Bbb{Q}^3$. In this basis the matrix $A$ of the multiplication by $\alpha$ is the companion matrix of its minimal polynomial and $B=2I-3A+2A^2$.