Exercise 14, § 3

Let A be a complete noetherian local ring, whose residue field is of characteristic $p > 0$. In the ring of formal series $A[[T]]$, one considers the elements $\omega_n = (1 - T)^{p^n}$ and $\gamma_n = 1 - \omega_n$ for every integer $n > 0$. Now prove that $\gamma_n$ is, up to sign, a distinguished polynomial (No. 8); deduce that $A_n = A[[T]]/(\gamma_n)$ identifies with the algebra over A of the group $G_n = \mathbf{Z}/p^n\mathbf{Z}$. Now prove that the intersection of the principal ideals $(\gamma_n)$ is reduced to 0; deduce that $A[[T]]$ identifies with the projective limit $\lim_{\leftarrow} A_n$.