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.