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- Jun 22, 2012

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

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

[TEX] p(x) = x^3 + 9x + 6 = x^3 + a_1 x + a_0 [/TEX]

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

and [TEX] a_1 = 9 \in (3) [/TEX]

and [TEX] a_0 = 6 \in (3) [/TEX] and [TEX] a_0 \notin (9) ([/TEX]

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

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

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

I would be grateful for some help with this problem.

Peter

Exercise 1 on page 519 reads as follows:

===============================================================================

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

===============================================================================

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

[TEX] p(x) = x^3 + 9x + 6 = x^3 + a_1 x + a_0 [/TEX]

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

and [TEX] a_1 = 9 \in (3) [/TEX]

and [TEX] a_0 = 6 \in (3) [/TEX] and [TEX] a_0 \notin (9) ([/TEX]

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

----------------------------------------------------------------------------------------------------------

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

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

I would be grateful for some help with this problem.

Peter

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