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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 )."===============================================================================
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
Exercise 1 on page 519 reads as follows:
===============================================================================
"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 )."===============================================================================
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]
----------------------------------------------------------------------------------------------------------
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
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