Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
8a. Show that if $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$, then $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$.
8b. Conversely, show that if $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$, then $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$.
8c. Deduce that if $\mu _ { 1 } , \ldots , \mu _ { d }$ are complex numbers and if $P ( X ) = \prod _ { i = 1 } ^ { d } \left( X - \mu _ { i } \right)$, then $P ( X ) \in \mathbb { Q } [ X ]$ if and only if $$\forall n \geqslant 1 , \quad \sum _ { i = 1 } ^ { d } \mu _ { i } ^ { n } \in \mathbb { Q }$$