# On cyclotomic field extensions

• Jul 20th 2011, 01:11 PM
costanza
On cyclotomic field extensions
This is part of a proof which characterises the Galois groups of cyclotomic field extensions, however this part shouldn't require any knowledge of such things. I'm stuck on another part of the proof too (it's a few pages long) but I'll deal with this bit first

Let $\zeta$ be the standard nth primitive root of unity and let p be some prime not dividing n. Let P be a maximal ideal of $\mathbb{Z}[\zeta]$ containing p. We have that $\zeta^{r}-1\in{P}$ for some r between 1 and n-1. The next line of the proof reads:

$(\zeta-1)(\zeta^{r-1}+...+\zeta+1)\equiv(\zeta-1)r$ $mod(P)$

It is this line that I don't understand. I feel it may have something to do with the fact that the second bracketed term has r additions but I'm unsure and would really appreciate any help.

Thanks

Paul