Let $z \in \mathbb { P } _ { n }$ and let $p$ be a prime number not dividing $n$. (a) Express as a function of $n$ the number $\prod _ { 1 \leq i < j \leq n } \left( z _ { i } - z _ { j } \right) ^ { 2 }$, where $z _ { 1 } , z _ { 2 } , \ldots , z _ { n }$ are the roots of the polynomial $P = X ^ { n } - 1$. Hint: One may consider the numbers $P ^ { \prime } \left( z _ { i } \right)$. (b) Show that $\Pi _ { z } \left( z ^ { p } \right) = 0$. Hint: show that if $\Pi _ { z } \left( z ^ { p } \right) \neq 0$, then there exists an algebraic integer $u$ such that $n ^ { n } = u \cdot \Pi _ { z } \left( z ^ { p } \right)$. (c) Conclude that $\Phi _ { n } = \Pi _ { z }$.
Let $z \in \mathbb { P } _ { n }$ and let $p$ be a prime number not dividing $n$.
(a) Express as a function of $n$ the number $\prod _ { 1 \leq i < j \leq n } \left( z _ { i } - z _ { j } \right) ^ { 2 }$, where $z _ { 1 } , z _ { 2 } , \ldots , z _ { n }$ are the roots of the polynomial $P = X ^ { n } - 1$.\\
Hint: One may consider the numbers $P ^ { \prime } \left( z _ { i } \right)$.\\
(b) Show that $\Pi _ { z } \left( z ^ { p } \right) = 0$.\\
Hint: show that if $\Pi _ { z } \left( z ^ { p } \right) \neq 0$, then there exists an algebraic integer $u$ such that $n ^ { n } = u \cdot \Pi _ { z } \left( z ^ { p } \right)$.\\
(c) Conclude that $\Phi _ { n } = \Pi _ { z }$.