The situation is the following: Let $f(x)= x^3-x+1\in\mathbb{Q}[x]$ and let $u$ be some root of $f$. Then define $b= -2u^2+u+1\in \mathbb{Q}(u)$. The claim is that $\mathbb{Q}(u) = \mathbb{Q}(b)$, and one should also find the minimal polynomial of $b$.
I was able to show the first part by looking at the degrees of the involved field extensions:
$$[\mathbb{Q}(u): \mathbb{Q}] = [\mathbb{Q}(u):\mathbb{Q}(b)] \cdot [\mathbb{Q}(b):\mathbb{Q}] $$
Then since $f$ is irreducible in $\mathbb{Q}$, the degree of $\mathbb{Q}(u)$ over $\mathbb{Q}$ cannot be one, and we know that it is smaller or equal to deg$f = 3$, so it must be $2$ or $3$. But these are both prime, so one of the degrees on the lhs must be equal to 1. Then one can show that the degree of $\mathbb{Q}(b)$ over $\mathbb{Q}$ cannot be one explicitly by noting that if that were the case, then $b\in \mathbb{Q}$, so there is an $a\in\mathbb{Q}$ sucht that
$$2u^2 - u+a = 0 $$
i.e. $u$ would be a root of $g(x) = 2x^2-x+a$ and then finding a contradiction to $u\not\in \mathbb{Q}$.
The problem is that I have no idea how to calculate the minimal polynomial of $b$ now, since I have done the first part so abstractly that I have basically no information on $u$ or $b$. I've been sitting on this problem for days now, so any help would be appriciated!
It is a bit, umm, cheesy, but you may mimic this with matrices, beginning with a "companion" matrix for $u^3 - u + 1$ as square $$ U = \left( \begin{array}{rrr} 0&1&0\\ 0&0&1\\ -1&1&0 \\ \end{array} \right) $$ for which, well $U^3 - U + I = 0$
Part 2: calculate $B = -2 U^2 + U + I.$ Which is all integers. I get: $$ B = \left( \begin{array}{rrr} 1&1&-2\\ 2&-1&1\\ -1&3&-1 \\ \end{array} \right) $$
$$ B^2 = \left( \begin{array}{rrr} 5&-6&1\\ -1&6&-6\\ 6&-7&6 \\ \end{array} \right) $$
Part 3: Find the characteristic polynomial of $B,$ which is monic cubic with integer coefficients. If there are repeat roots the minimal polynomial of $B$ is allowed to have lower degree.
Either way, what is the minimal polynomial of the square matrix $B??$