## Field Theory - Dummit and Foote Ch 13 - Exercise 1, page519

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