Exercise 17, § 1

Let A be the ring $\mathbf{Q}[X]$. Let $\Omega = (\Omega_n)_{n \in \mathbf{N}}$ be the element of A such that $\Phi_A(\Omega) = (X, X, ..., X, ...)$. For every element $a$ of $\mathbf{Z}_p$, put

a) For every $a \in \mathbf{Z}_p$, and every integer $n \in \mathbf{N}$, one has $\Omega_n(a) \in \mathbf{Z}_p$.

b) For every $a \in \mathbf{Z}$ and every integer $n \in \mathbf{N}$, one has $\Omega_n(a) \in \mathbf{Z}$, and, if $a \in \mathbf{N}$, $\Omega(a)$ is the element of $W(\mathbf{Z})$ equal to the sum of $a$ terms each equal to the unit element of $W(\mathbf{Z})$.

c) The mapping $a \mapsto \Omega(a)$ defines a homomorphism of $\mathbf{Z}_p$ into $W(\mathbf{Z}_p)$ which, when $\mathbf{Z}_p$ is identified with $W(F_p)$ (§ 1, No. 7, Example 3), coincides with the homomorphism $s_{F_p}$ defined in Exercise 15.