Exercise 9, § 2

Let A be a complete topological ring and $\mathfrak{m}$ a two-sided ideal of A. Suppose that the ring $A/\mathfrak{m}$ is a field, that every element of $\mathfrak{m}$ is topologically nilpotent (Gen. Top., III, §6, p. 317, Exercise 9), and that there exists a fundamental system of neighborhoods of 0 in A consisting of subgroups of the additive group of A. Then the ring A is local and $\mathfrak{m}$ is its maximal ideal. In particular, for every prime $p$, the ring $\mathbf{Z}_p$ of $p$-adic integers (V, §12, No. 3, p. 96) is local, and its maximal ideal is $p\mathbf{Z}_p$.