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let $\alpha = 2^\frac{1}{3}$ and $ \omega = e^{\frac{2\pi i}{3}} $

1) Assuming a, b, and c are rational numbers, compute and simplify the following expression as much as possible.

$(a + b\alpha + c\alpha^2)(a+b\omega\alpha+c\alpha^2\omega^2)(a+b\omega^2+c\omega\alpha^2) $

2) Prove that the set $\mathbb{Q}[\alpha] = $ {$a + b\alpha + c\alpha^2 | a, b, c \in \mathbb{Q} $} is closed under addition and multiplication.

3) Prove that $z \in \mathbb{Q}(\alpha)$ is non-zero then there exists $z^{-1} \in \mathbb{Q}(\alpha)$ such that $z*z^{-1} = 1 $ (hint: use part 1). Use parts 2 and 3 of the problem to prove that $\mathbb{Q}(\alpha)$ is a subfield of $\mathbb{C}$

I know part 1 is grueling -- I don't need a step by step simplification, but key steps would be helpful.

kt046172
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  • It may be grueling, but when I did it the first time, I found it instructive. My advice: don’t be afraid of pencil and paper. – Lubin Jan 22 '20 at 23:40
  • I find the 3rd factor of the product quite odd. There must be a mistake: $a+b\alpha \omega^2 + c\omega \alpha^2$ – Jean-Claude Colette Jan 23 '20 at 00:04

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