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Suppose $\alpha \in \Bbb{C}$ is zero of $x^3-x+1$. Express $(\alpha)^{-1},(\alpha+1)^{-1}, (\alpha^2+1)^{-1}$ in terms of $a_0+a_1\alpha+a_2\alpha^2$ for $a_i\in\Bbb{Q}$.

We know $Q[\alpha]\equiv Q[x]/\langle x^3-x+1\rangle$ which is a field.

So the above-mentioned should exist. We know $(\alpha)^{-1}$ can be expressed as $-\alpha^2+1$ and $(\alpha+1)^{-1}$ as $(-\alpha(\alpha-1))$. But I am not able to find $(\alpha^2+1)^{-1}$.

Also, is there a method to find it?

Bill Dubuque
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Raheel
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  • This is a straightforward simple mechanical computation using the extended Euclidean algorithm - see the linked dupe. – Bill Dubuque Jul 20 '24 at 20:26
  • We can also use inverse reciprocity to reduce it to computing inverse of complex integers = see here where I do this example that way. – Bill Dubuque Jul 20 '24 at 21:24

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