Dummit and Foote Chapter 13, Exercise 2, page 519 reads as follows:

"Show that $\displaystyle x^3 - 2x - 2 $ is irreducible over $\displaystyle \mathbb{Q} $ and let $\displaystyle \theta $ be a root.

Compute $\displaystyle (1 + \theta ) ( 1 + \theta + {\theta}^2) $ and $\displaystyle \frac{(1 + \theta )}{ ( 1 + \theta + {\theta}^2)} $ in $\displaystyle \mathbb{Q} (\theta)$

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My attempt at this problem so far is as follows:

$\displaystyle p(x) = x^3 - 2x - 2 $ is irreducible over $\displaystyle \mathbb{Q} $ by Eisenstein's Criterion.

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

Proceeding this way one finds the following:

$\displaystyle (1 + \theta ) ( 1 + \theta + {\theta}^2) = 1 + 2{\theta} + 2{\theta}^2 + {\theta}^3 $

$\displaystyle = ({\theta}^3 - 2{\theta} - 2) + (2{\theta}^2 + 4{\theta} + 3) $

$\displaystyle 2{\theta}^2 + 4{\theta} + 3 $

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