Let $z \in \mathbb { P } _ { n }$ and let $p$ be a prime number not dividing $n$. (a) Let $F , G \in \mathbb { Z } [ X ]$. Show that there exists $H \in \mathbb { Z } [ X ]$ such that $$( F + G ) ^ { p } = F ^ { p } + G ^ { p } + p H$$ (b) Show that $\Pi _ { z } \in \mathbb { Z } [ X ]$ and deduce the existence of a polynomial $F \in \mathbb { Z } [ X ]$ such that $$\Pi _ { z } \left( X ^ { p } \right) = \Pi _ { z } ( X ) ^ { p } + p F ( X )$$ (c) Show that $\frac { \Pi _ { z } \left( z ^ { p } \right) } { p }$ is an algebraic integer.
Let $z \in \mathbb { P } _ { n }$ and let $p$ be a prime number not dividing $n$.
(a) Let $F , G \in \mathbb { Z } [ X ]$. Show that there exists $H \in \mathbb { Z } [ X ]$ such that
$$( F + G ) ^ { p } = F ^ { p } + G ^ { p } + p H$$
(b) Show that $\Pi _ { z } \in \mathbb { Z } [ X ]$ and deduce the existence of a polynomial $F \in \mathbb { Z } [ X ]$ such that
$$\Pi _ { z } \left( X ^ { p } \right) = \Pi _ { z } ( X ) ^ { p } + p F ( X )$$
(c) Show that $\frac { \Pi _ { z } \left( z ^ { p } \right) } { p }$ is an algebraic integer.