Originally Posted by

**Aprshaw** Hey all, I am having a problem with an exercise from the book "A Classical Introduction to Modern Number Theory". It involves the ring $\displaystyle \mathbb{Z}[\omega]$, where

$\displaystyle

\omega = \frac{-1 + \sqrt{-3}}{2}

$

The question is

For any $\displaystyle \alpha \in \mathbb{Z}[\omega]$, show that $\displaystyle \alpha$ is congruent to 1, -1 or 0 mod 1 - $\displaystyle \omega$

I figured the best thing to use was Euclidean division, so I set

$\displaystyle \alpha = a + b\omega$

and worked out that

$\displaystyle \frac{\alpha}{1 - \omega} = \frac{2a - b + (a + b)\omega}{3}$

I can see how it would work, and have tested it a few times on random Eisenstein integers, but cannot work out how to actually prove it!

Any help would be much appreciated