Exercise 13, § 2

Let $A$ be a complete discrete valuation ring of characteristic zero, whose residue field is of characteristic $p$. Let $K$ be the field of fractions of $A$, $\hat{K}$ an algebraic closure of $K$, provided with the unique valuation $v$ extending that of $K$ (VI, § 8, No. 7), $\hat{K}$ the completion of $K$, $\hat{A}$ the valuation ring of $\hat{K}$. Put $e = v(p)$.

a) The series $\log(1 + x) = \sum_{i=1}^\infty (-1)^{i-1} x^i / i$ converges for $v(x) > 0$ and defines homomorphisms of $\mathbf{Z}_p$-modules (cf. Exercise 12)

b) The series $\exp(x) = \sum_{i=0}^\infty x^i / i!$ converges on the ideal $a$ of $\hat{K}$ consisting of elements of valuation $> e/(p-1)$, and defines homomorphisms of $\mathbf{Z}_p$-modules

c) For $x \in a$, one has $\log(\exp(x)) = x$, $\log(1 + x) \in a$ and $\exp(\log(1 + x)) = 1 + x$. Thus exp defines isomorphisms of $\mathbf{Z}_p$-modules

with inverses defined by log.

d) The kernel of $\log$ on $1 + m_{\hat{A}}$ consists of the roots of unity whose order is a power of $p$. The image $\log(1 + m_{\hat{A}})$ is equal to $m_{\hat{A}}$.