# Thread: Primitive pth root of unity of a prime p

1. ## Primitive pth root of unity of a prime p

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

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?

b) Show that $\displaystyle disc( \omega ) = -p^{p-2}$, in addition, $\displaystyle disc ( \omega ) = p^ {p-2}$ iff $\displaystyle p \equiv 1 \ (mod \ 4)$

I just can't really get a hand on disc, any hint?

Thank you!

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?
I would do it a little differently. The mimimal polynomial is $\displaystyle 1+x+...+x^{p-1}$. It happens to be irreducible so that is why it is minimal (you never said why you poynomial was minimal). And then differenciate term-by-term.

b) Show that $\displaystyle disc( \omega ) = -p^{p-2}$, in addition, $\displaystyle disc ( \omega ) = p^ {p-2}$ iff $\displaystyle p \equiv 1 \ (mod \ 4)$

I just can't really get a hand on disc, any hint?
I never seen "disk" before.

3. Disc is the discriminant, so is the determinant of $\displaystyle \{ \omega ^{1}, \omega ^{2}, . . . , \omega ^{p-1} \}$