习题 14, § 3

Let $A$ be a complete 诺特局部环 whose 剩余类域 is of characteristic $p > 0$. In 环 of formal 幂级数 $A[[T]]$ consider 元素 $\omega_n = (1 - T)^{p^n}$ and $\gamma_n = 1 - \omega_n$ for every 整数 $n > 0$. Show that $\gamma_n$ is, to within a sign, a distinguished polynomial (no. 8); deduce that $A_* = A[[T]]/(\gamma_n)$ is identified with 代数 over $A$ of 群

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