Let $P \in \mathbb { Z } [ X ]$ be a monic polynomial of degree $n \geq 1$, irreducible in $\mathbb { Q } [ X ]$ and all of whose complex roots have modulus 1. Let $z _ { 1 } , \ldots , z _ { n }$ be the complex roots of $P$ counted with their multiplicities, so that $P = \prod _ { i = 1 } ^ { n } \left( X - z _ { i } \right)$. For every integer $k \geq 0$ we denote $a _ { k } = z _ { 1 } ^ { k } + z _ { 2 } ^ { k } + \cdots + z _ { n } ^ { k }$. (a) Show that the series $\sum _ { k \geq 0 } a _ { k } z ^ { k }$ converges for all $z \in \mathbb { C }$ such that $| z | < 1$. (b) Let $z \in \mathbb { C }$ be non-zero such that $| z | < 1$ and let $f ( z )$ be the sum of the series $\sum _ { k \geq 0 } a _ { k } z ^ { k }$. Show that $$z f ( z ) P \left( \frac { 1 } { z } \right) = P ^ { \prime } \left( \frac { 1 } { z } \right)$$ (c) Deduce that $a _ { k } \in \mathbb { Z }$ for all $k \geq 0$.
Let $P \in \mathbb { Z } [ X ]$ be a monic polynomial of degree $n \geq 1$, irreducible in $\mathbb { Q } [ X ]$ and all of whose complex roots have modulus 1. Let $z _ { 1 } , \ldots , z _ { n }$ be the complex roots of $P$ counted with their multiplicities, so that $P = \prod _ { i = 1 } ^ { n } \left( X - z _ { i } \right)$. For every integer $k \geq 0$ we denote $a _ { k } = z _ { 1 } ^ { k } + z _ { 2 } ^ { k } + \cdots + z _ { n } ^ { k }$.
(a) Show that the series $\sum _ { k \geq 0 } a _ { k } z ^ { k }$ converges for all $z \in \mathbb { C }$ such that $| z | < 1$.\\
(b) Let $z \in \mathbb { C }$ be non-zero such that $| z | < 1$ and let $f ( z )$ be the sum of the series $\sum _ { k \geq 0 } a _ { k } z ^ { k }$. Show that
$$z f ( z ) P \left( \frac { 1 } { z } \right) = P ^ { \prime } \left( \frac { 1 } { z } \right)$$
(c) Deduce that $a _ { k } \in \mathbb { Z }$ for all $k \geq 0$.