Suppose that $\alpha\in \mathbb{R}$ is the root of a polynomial $p(X)$. Let $g(X)$ be a polynomial such that $f_{\mathbb{Q}}^{g(\alpha)}$ is the minimal polynomial of $g(\alpha)$ over $\mathbb{Q}$. Is there a general way for finding the minimal polynomial $f_{\mathbb{Q}}^{1/g(\alpha)}$ of $\frac{1}{g(\alpha)} \in \mathbb{R}$ over $\mathbb{Q}$.
In particular, I have $\alpha$ to be the root of $p(X) = X^3 - X - 1$. Furthermore, let $g(X) = X^2 + 1$, than $f_{\mathbb{Q}}^{\alpha^2 + 1} = X^3 - 5X^2 + 8X - 5$ (see https://math.stackexchange.com/a/2223606/714378). I would like to find the minimal polynomial of $1/(\alpha^2+1)$. Is there a general/clever way to encounter this problem without a lot of calculation?