I am studying Dummit and Foote Chapter 13: Field Theory.

Exercise 1 on page 519 reads as follows:

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"Show that $\displaystyle p(x) = x^3 + 9x + 6 $ is irreducible in $\displaystyle \mathbb{Q}[x] $. Let $\displaystyle \theta $ be a root of p(x). Find the inverse of $\displaystyle 1 + \theta $ in $\displaystyle \mathbb{Q} ( \theta ) $."


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Now to show that $\displaystyle p(x) = x^3 + 9x + 6 $ is irreducible in $\displaystyle \mathbb{Q}[x] $ use Eisenstein's Criterion

$\displaystyle p(x) = x^3 + 9x + 6 = x^3 + a_1 x + a_0 $

Now (3) is a prime ideal in the integral domain $\displaystyle \mathbb{Q} $

and $\displaystyle a_1 = 9 \in (3) $

and $\displaystyle a_0 = 6 \in (3) $ and $\displaystyle a_0 \notin (9) $

Thus by Eisenstein, p(x) is irreducible in $\displaystyle \mathbb{Q}[x] $

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However, I am not sure how to go about part two of the problem, namely:

"Let $\displaystyle \theta $ be a root of p(x). Find the inverse of $\displaystyle 1 + \theta $ in $\displaystyle \mathbb{Q} ( \theta ) $."

I would be grateful for some help with this problem.

Peter