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

Exercise 1 on page 519 reads as follows:

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"Show that is irreducible in . Let be a root of p(x). Find the inverse of in ."

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Now to show that is irreducible in use Eisenstein's Criterion

Now (3) is a prime ideal in the integral domain

and

and and

Thus by Eisenstein, p(x) is irreducible in

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

"Let be a root of p(x). Find the inverse of in ."

I would be grateful for some help with this problem.

Peter