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**tcRom** Let $\displaystyle \omega = e^{(2\pi i)/p^2}$ and $\displaystyle f(x)= x^{p^2} - 1 $ where p is a prime > 2.

Show that $\displaystyle \omega$ is a root of f(x). done...

Show that $\displaystyle (x^p - 1)$ is a factor of f(x). done...

I used geometric series to extrapolate f(x)

$\displaystyle = x^{p^2} - 1$

$\displaystyle = (x^p)^p - 1$

$\displaystyle = u^p - 1$

$\displaystyle = (u-1)(1 + u + u^2 + u^3 + ... + u^{p - 1})$

$\displaystyle = (x^p - 1)(1 + x^p + x^2p + x^3p + ... + x^{p^2 - p})$

Now, I am having trouble showing that $\displaystyle \omega$ is a root of g(x) for $\displaystyle f(x)=(x^p - 1)g(x)$. When I just try to plug and play in g(x), I don't get anything that heads towards 0. Step by step or a skeleton is fine with me, I just need a bit of a push.