Exercise 14, § 3

Let $A$ be a complete Noetherian local ring whose residue field is of characteristic $p > 0$. In the ring of formal power series $A[[T]]$ consider the elements $\omega_n = (1 - T)^{p^n}$ and $\gamma_n = 1 - \omega_n$ for every integer $n > 0$. Show that $\gamma_n$ is, to within a sign, a distinguished polynomial (no. 8); deduce that $A_* = A[[T]]/(\gamma_n)$ is identified with the algebra over $A$ of the group

$G_n = \mathbf{Z}/p^n\mathbf{Z}$. Show that the intersection of the principal ideals $(\gamma_n)$ is reduced to 0; deduce that $A[[T]]$ is identified with the inverse limit $\varprojlim A_*.$