Constant Split Embedding Problems over Complete Fields

Abstract

Let K ₀ be a complete field under a discrete ultrametric absolute value and x an indeterminate. We prove that each finite split embedding problem over K ₀ has a rational solution. Thus, given a finite Galois extension K of K ₀ with Galois group Γ that acts on a finite group G, there is a finite Galois extension F of K ₀(x) which contains K(x) with Gal(F/K(x))≅G and Gal(F/K ₀(x))≅Γ⋉G such that res: Gal(F/K ₀(x))→Gal(K/K ₀) corresponds to the projection Γ⋉G→Γ. Moreover, F has a K-rational place unramified over K(x) whose decomposition group over K ₀(x) is Γ.

To construct F we choose finitely many cyclic subgroups C _i , i∈I, of G which generate G. For each i∈I we construct a Galois extension F _i =K(x,z _i ) of K(x) with Galois group C _i in K((x)). Then we consider the ring R=K{w _i |i∈I} as in Section 3.2, where $w_{i}={r\over x-c_{i}}$, r∈K ₀, c _i ∈K, and |r|≤|c _i −c _j | for all i≠j, and shift F _i into the field P′ _i =Quot(K{w _i }) (Lemma 4.3.5). Choosing the c _i ’s in an appropriate way (Claim A of the proof of Proposition 4.4.2), we establish patching data $\mathcal{E}$ with a proper action of Γ and apply Proposition 1.2.2 to solve the given embedding problem.