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Given a root $\alpha \in \bar{\mathbb{Q}}$ of the polynomial $X^3+X+1$, how do I find the mimimal polynomial for the element $\alpha^{-1}$?

Research effort

The polynomial $X^3+X+1$ is irreducible. The set $\{f: f(\alpha)=0\}$ is a principal ideal. If this is generated by $g$, than $g,f$ differ a unit, so we can say that the principal ideal is generated by f. Now I try to find a polynomial $h \in \mathbb{Q}[X]$ such that $h(\alpha^{-1})=0$, and I have no idea how i should do that. can you give me a hint?

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  • $\begingroup$ Multiply $\alpha^3+\alpha+1=0$ by $\alpha^{-3}$. $\endgroup$ – Derek Holt Nov 16 '13 at 21:34
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Hint: since $\mathbb{Q}[\alpha^{-1}]=\mathbb{Q}[\alpha]$ they must have the same degree; divide the equality $\alpha^3+\alpha+1=0$ by …

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