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**tttcomrader** Let $\displaystyle K= \mathbb {Q}[ \omega ] $, where $\displaystyle \omega $ is a primitive pth root of unity for an odd prime p, and write N for $\displaystyle N_{K}$.

a) Let f be the minimum polynomial of $\displaystyle \omega $. Show that [tex] f'( \omega ) = \frac {p}{ \omega ( \omega - 1)}

proof so far:

Now, $\displaystyle x^p - 1 = (x-1)f(x) $

Then $\displaystyle px^{p-1} = (x-1)f'(x) + f(x) $

Plug in $\displaystyle \omega $, then $\displaystyle f( \omega ) =1 $, then we have:

$\displaystyle p \omega ^{p-1} = ( \omega - 1) f'( \omega) + 1 $

$\displaystyle \frac {p \omega ^{p-1} - 1} { \omega - 1 } = f' ( \omega ) $

Am I doing this right so far?