I am currently stuck on a problem and I would like a hint as to what direction I should go in.
Let $\alpha$ be a root of $x^3-x-1$. I am asked to find the degrees of the extensions $\mathbb{Q}[\alpha]$, $\mathbb{Q}[\alpha +1]$, and $\mathbb{Q}[\alpha^2]$, as well as the minimal polynomials of $\alpha, \alpha+1, \alpha^2$. I know that reducing $x^3-x-1 \equiv 0\mod{2}$ gives us the congruence $-1 \equiv 0\mod{2}$ and thus the polynomial is irreducible making the degree of $\mathbb{Q}[\alpha]= 3$.
From this I can deduce that $\langle 1, \alpha, \alpha^2 \rangle$ form a basis for the extension, and I can see that both of $\alpha+1, \alpha^2 \in \mathbb{Q}[\alpha]$. Given that the extension is finite and $\alpha+1, \alpha^2 \in \mathbb{Q}[\alpha]$ I know that the degrees of $\mathbb{Q}[\alpha +1]$ and $\mathbb{Q}[\alpha^2]$ must be $\leq 3$, and since $\alpha \notin \mathbb{Q}$ their degrees must also be greater than $1$.
My current thoughts are that we might have that $\mathbb{Q}[\alpha+1]=\mathbb{Q}[\alpha]$ similar to how $\mathbb{Q}[\sqrt[]{2}+1]=\mathbb{Q}[\sqrt[]{2}]$, but I don't think that this always holds.
At this point I am currently stuck, and would appreciate any help in the right direction, including how to calculate the minimal polynomial for some arbitrary algebraic element.
Thanks