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- Jun 22, 2012

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"Show that [TEX] x^3 - 2x - 2 [/TEX] is irreducible over [TEX] \mathbb{Q} [/TEX] and let [TEX] \theta [/TEX] be a root.

Compute [TEX] (1 + \theta ) ( 1 + \theta + {\theta}^2) [/TEX] and [TEX] \frac{(1 + \theta )}{ ( 1 + \theta + {\theta}^2)} [/TEX] in [TEX] \mathbb{Q} (\theta)[/TEX]

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My attempt at this problem so far is as follows:

[TEX] p(x) = x^3 - 2x - 2 [/TEX] is irreducible over [TEX] \mathbb{Q} [/TEX] by Eisenstein's Criterion.

To compute [TEX] (1 + \theta ) ( 1 + \theta + {\theta}^2) [/TEX] I adopted the simple (but moderately ineffective) strategy of multiplying out and trying to use the fact that [TEX] \theta [/TEX] is a root of p(x) - that is to use the fact that [TEX] {\theta}^3 - 2{\theta} - 2 = 0 [/TEX].

Proceeding this way one finds the following:

[TEX] (1 + \theta ) ( 1 + \theta + {\theta}^2) = 1 + 2{\theta} + 2{\theta}^2 + {\theta}^3 [/TEX]

[TEX] = ({\theta}^3 - 2{\theta} - 2) + (2{\theta}^2 + 4{\theta} + 3) [/TEX]

[TEX] 2{\theta}^2 + 4{\theta} + 3 [/TEX]

Well, that does not seem to be going anywhere really! I must be missing something!

Can someone please help with the above and also help with the second part of the question ...

Peter

[Note: The above has also been posted on MHF]