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I'm reading a paper in which they solve the equation: $$a^3-2b^3=\pm 1$$ in integers using algebraic number theory.

The number $a-b\alpha$, with $\alpha=\sqrt[3]{2}$, is a unit in $\mathbb{Z}[\alpha]$. The units of this ring are, up to sign, powers of the single unit $1+\alpha+\alpha^2$. With some work one finds that $|a-b\alpha|$ can only be the zero'th power, so that $a=\pm 1$ and $b=0$.

1) Why are the units of the ring only powers of $1+\alpha+\alpha^2$? I can't find anything to this effect in my textbook and web searches won't turn up with anything.

2) Can't $|a-b\alpha|$ be the $(-1)$th power as well and $a=b=\pm 1$?

The second question I've concluded is a minor error but I can't be satisfied with this solution without a proof for my first question.

The Paper in question: http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf

Relevant section is towards the end of second page.

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    I think they might be using Dirchlet's unit theorem here, and also showing that $1 + \alpha + \alpha^2$ is the fundamental unit. Check Dirichlet's unit theorem on the web.2012-11-28
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    Dirichlet's unit theorem definitely implies the group of units is isomorphic to $\mathbb{Z}$ because $\mathbb{Q}(2^{1/3})$ has one real embedding and 2 complex embeddings into $\mathbb{C}$.2012-11-28
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    Thanks for giving me something to go with. I'm about to look up the theorem now.2012-11-28
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    Dirichlet's unit theorem is one of the most beautiful theorems in algebraic number theory. It lets you describe the structure of the group of units of nice rings in terms of geometry.2012-11-28

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