6. a. For $q \in \mathbb{N}$, we set $S_q = \sum_{\ell=0}^{p-1} \ell^q$. Observe that $p$ divides $\sum_{\ell=0}^{p-1} ((\ell+1)^{q+1} - \ell^{q+1})$ and deduce by recursion that $p$ divides $S_q$ for $0 \leq q \leq p-2$. b. Let $Z = [z_{ij}]$ and $Z' = [z'_{ij}]$ be two square matrices of order $N$ with entries in $\mathbb{Z}$. We define the relation $Z \equiv Z'[p]$ by $z_{ij} \equiv z'_{ij}[p]$ for $1 \leq i, j \leq N$. Prove that $$(M_{p-1})^{(p-1)} \equiv (-1)^{(p-1)/2} \mathrm{Id} \quad [p].$$ c. What can be said about a polynomial $Q$ with integer coefficients such that $Q(M_{p-1}) \equiv 0[p]$?
6. a. For $q \in \mathbb{N}$, we set $S_q = \sum_{\ell=0}^{p-1} \ell^q$. Observe that $p$ divides $\sum_{\ell=0}^{p-1} ((\ell+1)^{q+1} - \ell^{q+1})$ and deduce by recursion that $p$ divides $S_q$ for $0 \leq q \leq p-2$.\\
b. Let $Z = [z_{ij}]$ and $Z' = [z'_{ij}]$ be two square matrices of order $N$ with entries in $\mathbb{Z}$. We define the relation $Z \equiv Z'[p]$ by $z_{ij} \equiv z'_{ij}[p]$ for $1 \leq i, j \leq N$. Prove that
$$(M_{p-1})^{(p-1)} \equiv (-1)^{(p-1)/2} \mathrm{Id} \quad [p].$$
c. What can be said about a polynomial $Q$ with integer coefficients such that $Q(M_{p-1}) \equiv 0[p]$?