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

Exercise 1 on page 519 reads as follows:

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


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

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

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

and  a_1 = 9 \in (3)

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

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

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

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

I would be grateful for some help with this problem.

Peter