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