Show that if $f$ is cyclic, then $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free in $\mathcal{L}(E)$ and the minimal polynomial of $f$ has degree $n$.

Show that if $f$ is cyclic, then $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free in $\mathcal{L}(E)$ and the minimal polynomial of $f$ has degree $n$.

Let $a \in \mathbb{K}$. For all $p \in \mathbb{K}[X]$, we set $E_a(p) = E_a p = p(X+a)$.
Show that $E_a$ is an automorphism of $\mathbb{K}[X]$.

a) Show that the restriction map to $\mathbb{H}^{\mathrm{im}}$ induces a group isomorphism $$\mathrm{Aut}(\mathbb{H}) \simeq \mathrm{SO}(\mathbb{H}^{\mathrm{im}}).$$ b) Show that $$\mathrm{Aut}(\mathbb{H}) = \{\alpha(u,u) \mid u \in S\}.$$