On Grunwald-Hasse-Wang's theorem
Hiroo Miki · Journal of the Mathematical Society of Japan · 1978
Let $k$ be a field with discrete valuations, $k_{\mathfrak{p}}$ the completion of $k$ with respect to a prime divisor $\mathfrak{p}$ of $k,$ $k_{sep}$ the separable algebraic closure of $k$ , and $k_{\mathfrak{p}_{sep}}$ , the separable algebraic closure of $k_{\mathfrak{p}}$ containing $k_{sep}$ .Here a prime divisor is a normalized discrete valuation.Let $S$ be a finite set of prime divisors of $k$ , $G$ a finite abelian group, and $(K^{\mathfrak{p}}, j_{\mathfrak{p}})$ a pair of a finite abelian extension $K^{\mathfrak{p}}$ of $k_{\mathfrak{p}}$ in $k_{\mathfrak{p}_{sep}}$ , and an injective homomorphism $j_{\mathfrak{p}}$ from the Galois groupK\mathfrak{p}=K^{\mathfrak{p}}$ and $j_{\mathfrak{p}}=j\circ{\rm Res}_{\rho}$ for any $\mathfrak{p}\in S$ , where $K_{\mathfrak{p}}=Kk_{\mathfrak{p}}$ and ${\rm Res}_{p}$ : $G(K_{p}/k_{p})\rightarrow G(K/k)$ is the restriction from $K_{\mathfrak{p}}$ to $K$ .We call the pair $(K, j)$ a solution of the imbedding problem $P$ .When $k$ is a finite algebraic number field or an algebraic function field in one variable over a finite constant field, Grunwald, Hasse and Wang ([2], [3], [6]) gave a condition for an imbedding problem $P$ to have a solution, and in particular proved that an imbedding problem $P$ has a solution if $(k, G, S)$ is not the "special case" (see also Chap. 10 of [1] and Theorem of [5]).Their proofs were based on class field theory, and Hasse ([3], \S 4, 1) raised the problem of giving a proof based on Kummer theory.In the present paper, we shall give a certain sufficient condition for an imbedding problem $P$ to have a solution for any field $k$ with discrete valuations.More precisely, we shall prove the following THEOREM.Let $k$ be a field, $S$ a finite set of Prime divisors of $k$ , and $G$ a finite abelian group of tyPe $(p_{1}^{m_{1}}, p_{2}^{m_{2}}, \cdots, p_{t}^{m_{\zeta}})$ .Then an imbedding Problem $P\{k,$ $G,$ $S$ , $(F, j_{P})(\mathfrak{p}\in S)\}$ has a solution if the following two conditions are satisfied: (i) There exist $t$ distinct prime divisors $q_{1},$ $q_{2},$ $q_{t}$ of $k$ outside $S$ such that $\zeta(p_{\ell}^{m_{i}})\in k_{\mathfrak{q}_{i}}$ if $p_{i} eq ch(k)$ .(ii) If exp $(G)$ is divisible by 4 and if $ch(k) eq 2$ , then $\zeta(4)\in k$ (see Notation below and Theorem 5).