Find $c_1,c_2,c_3\in\mathbb{Q}$ such that $(1+\alpha^4)^{-1}=c_1+c_2\alpha+c_3\alpha^2$ in $\Bbb Q(\alpha)$. Let $\alpha\in \overline{\mathbb{Q}}$ be a root of $X^3+X+1\in\mathbb{Q}[X]$. So this is the minimal polynomial of $\alpha$ because it's irreducible in $\mathbb{Q}[X]$. I had to find the minimal polynomials of $\alpha^{-1}$ and $\alpha-1$. After that I have to do what the title says. I found the minimal polynomials $f_1=X^3+3X^2+4X+4$ and $f_2=X^3+X^2+1$ of $\alpha^{-1}$ and $\alpha-1$, respectively. Now I suppose I have to use this result to solve the question, but I don't have any idea how to this. I need help. Thanks.

Hint
Apply the extended Euclidean algorithm to find polynomals $g,h$ with
$$
g(X^4 + 1) + h(X^3 + X + 1) = 1.
$$
Then consider this equation mod $X^3 + X + 1$ and plug in $\alpha$.
Note: It's not strictly neccesary, but you may want to reduce $X^4 + 1$ mod $X^3 + X + 1$ first.
