Exercise 11, § 1

Let $A$ be a ring of characteristic $p$. a) Let $n \in \mathbf{N}$; prove that $p^{n-1}/n! \in \mathbf{Z}_{(p)}$. Prove that $(nm)!/n!(m!)^n$ is an integer for $n, m$ in $\mathbf{N}$. b) For $n \in \mathbf{N}$, let $\gamma_n : V_1(A) \to W(A)$ be the mapping defined by

(i) Prove that one has $\gamma_n(x) \in V_1(A)$ if $n \geqslant 1$ and $x \in V_1(A)$. (ii) For $x, y \in V_1(A)$ and $n \in \mathbf{N}$, one has (iii) For $\lambda \in W(A), \ x \in V_1(A), \ n \in \mathbf{N}$, one has (iv) For $x \in V_1(A)$ and $n, m$ in $\mathbf{N}$, one has and (v) One has $F \gamma_n(Vx) = (p^n/n!) x^n$ and $\gamma_n(px) = (p^n/n!) x^n$, for $x \in W(A)$.