Exercise 16, § 2

Let A be a complete discrete valuation ring, with residue field $k$. a) Prove that there exists a multiplicative section $\varphi : k^* \to A$. (If A contains a field, use § 3. Otherwise, use Exercise 11, p. 72.) b) Let t be a uniformizing element of A, $\mathscr{C}$ the group generated by t, and $\varphi : k^* \to A$ a multiplicative section. The multiplicative group of the field of fractions of A decomposes into a direct product $\mathscr{C} \times \varphi(k^*) \times (1 + m_A)$. c) Suppose that A has equal characteristics and let K be a field of representatives. Then A is identified with the ring $K[[T]]$ of formal power series with coefficients in K, and the group $1 + m_A$ is identified with the group $\Lambda(K) = 1 + TK[[T]]$ (cf. p. 56, Exercise 42). If K is of characteristic zero, the mapping $\varphi \mapsto \frac{1}{T} \log \varphi$ of the group $\Lambda(K)$ into the additive group $K[[T]]$ is an isomorphism. If A is of characteristic $p$, $1 + m_A$ is identified with the group $W(A)^L$ where L is the set of positive integers prime to $p$ (use Exercises 40, h) p. 54 and 43, b), p. 57).