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If $\alpha \in \mathbb{Z}(\omega)$, show that $\alpha$ is congruent to either $0, 1$ or $-1$ modulo $1-\omega$.

Exercise 1 page 134 in the book 'A Classical Introduction to Modern Number Theory' of K. Ireland and M. Rosen.

Thanks a lot.

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$\omega$ being a primitive cubic root of unit? –  lhf Apr 30 '12 at 14:05
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and it's $\mathbb{Z}[\omega]$. –  lhf Apr 30 '12 at 14:07
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What have you tried? If this is homework, tag it as such. –  lhf Apr 30 '12 at 14:13
    
For the OP: using $\mathbb{Z}[\omega]$ instead of $\mathbb{Z}(\omega)$ is important since the latter suggests a field. Specifically when using a field, $\mathbb{F}[x]$ is standard notation for the polynomial ring, and $\mathbb{F}(x)$ is used for the field of "rational functions". –  rschwieb Apr 30 '12 at 15:06
    
Try and draw the 2D grid on the plane with the numbers of $\mathbb{Z}[\omega]$ marked with dots. Circle the ones that are congruent to zero modulo $1-\omega$. All will be clear (and this is a useful exercise in its own right)! –  Jyrki Lahtonen Apr 30 '12 at 21:11

1 Answer 1

Hint $\rm\: \omega\equiv 1\:\Rightarrow\: 0 = \omega^2\!+\!\omega\!+\!1\equiv 3\ $ so $\rm\ \omega\equiv 1,\ 3\equiv 0\:\Rightarrow\: m\!+\!n\omega \equiv (m\!+\!n)\:\! mod\ 3\equiv 0,\pm1$

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