Exercise 17, § 2

Let A be a complete discrete valuation ring, whose residue field $k$ is perfect of characteristic $p$ and whose field of fractions is of characteristic 0. Let $f$ be the unique homomorphism from $W(k)$ into A that induces the identity on the residue field $k$. Let finally $\pi$ be a uniformizer of A.

a) There exists an Eisenstein polynomial P with coefficients in $f(W(k))$ such that $P(\pi) = 0$ (use VIII, § 5, No. 4). b) Let e be the valuation of p. For every integer $n \geq 0$, denote by $U_n$ the multiplicative group $1 + m_A$, and put $\lambda(n) = \inf(np, n + e)$, $e_1 = e/(p - 1)$. Let u be the class in $k^*$ of $p\pi^{-e}$. Let $\varphi$ be the endomorphism of $U_1$ which associates $x^p$ to x. We then have, for every integer $n \geq 1$, $\varphi(U_n) \subset U_{\lambda(n)}$, $\varphi(U_{n+1}) \subset U_{\lambda(n)+1}$ and the induced homomorphism $\varphi_n : U_n/U_{n+1} \to U_{\lambda(n)}/U_{\lambda(n)+1}$ is an isomorphism if $n \neq e_1$. If $n = e_1$, $\varphi_n$ is injective if and only if the equation $x^{p-1} + u = 0$ has no solution in $k$, and surjective if and only if the equation $x^p + ux = y$ has a solution in $k$ for every $y \in k$. c) The following two conditions are equivalent: (i) A contains p p-th roots of unity; (ii) $e_1$ is an integer and the equation $x^{p-1} + u = 0$ has a solution in $k$. d) The topology induced on $U_1$ by that of A coincides with the p-adic topology of $U_1$. e) Let S be the set of integers $n \geq 1$ such that A contains $p^n$ $p^n$-th roots of unity. Then $s = \mathrm{Card}(S)$ is finite. f) Let I be the set of integers i prime to p and satisfying $1 \leq i < e + e_1$. Then we have $\mathrm{Card}(I) = e$ and the image of $\mathbf{N} - \{0\}$ by $\lambda$ is $\mathbf{N} - (I \cup \{0\})$. g) If $e_1$ is an integer and $\varphi_{e_1}$ is not surjective, put $I' = I \cup \{e + e_1\}$; otherwise, put $I' = I$. For each integer $i \in I'$, let $\pi_i$ be an element of A of valuation i. Then the mapping from $W(k)^{I'}$ into $U_1$ which associates $\prod_{i \in I'} (1 + \pi_i)^{a_i}$ to $(a_i)_{i \in I'}$ is surjective; it is a homomorphism of $Z_p$-modules. h) Let $n$ be an integer $> e_1$ and let $I(n)$ be the interval $[n, \lambda(n) - 1]$ of $\mathbf{N}$. For every integer $i \in I(n)$, let $\pi_i$ be an element of A of valuation i. Then the mapping from $W(k)^{I(n)}$ into $U_n$ which associates $\prod_{i \in I(n)} (1 + \pi_i)^{a_i}$ to $(a_i)_{i \in I(n)}$ is an isomorphism of $Z_p$-modules. i) Suppose that k is finite and denote by d the degree over $\mathbf{Q}_p$ of the field of fractions of A. Then the $Z_p$-module $1 + m_A$ is isomorphic to $(\mathbf{Z}/p^s\mathbf{Z}) \times \mathbf{Z}_p^d$. (One may use the preceding questions, or else use exerc. 13, p. 73.)