Until the end of part B, we assume that no root of $p$ is stable. For every $j \in \llbracket 1, n \rrbracket$, we define the rational function $g_j \in E$ by $$g_j = \frac{f_j}{\prod_{i=1}^{n}(1 - \alpha_i X)}$$ and the map $P_j$, which associates to a rational function $f \in E$ the rational function $$P_j(f) = \frac{(1 - \alpha_j X)f - (1 - \alpha_j^2)f(\alpha_j)}{X - \alpha_j}$$ Show that for every $j \in \llbracket 1, n \rrbracket$, the map $P_j$ is an endomorphism of $E$ and determine its kernel.
Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$, we define the rational function $g_j \in E$ by
$$g_j = \frac{f_j}{\prod_{i=1}^{n}(1 - \alpha_i X)}$$
and the map $P_j$, which associates to a rational function $f \in E$ the rational function
$$P_j(f) = \frac{(1 - \alpha_j X)f - (1 - \alpha_j^2)f(\alpha_j)}{X - \alpha_j}$$
Show that for every $j \in \llbracket 1, n \rrbracket$, the map $P_j$ is an endomorphism of $E$ and determine its kernel.