Split Embedding Problems over Complete Fields

Abstract

Let K ₀ be a complete field with respect to an ultrametric absolute value. In Proposition 4.4.2 we considered a finite Galois extension K of K ₀ with Galois group Γ acting on a finite group G and let x be an indeterminate. We constructed a finite Galois extension F of K ₀(x) that contains K and with Galois group Γ⋉G that solves the constant embedding problem Γ⋉G→Gal(K(x)/K ₀(x)). Using an appropriate specialization we have been then able to prove the same result in the case where K ₀ was an arbitrary ample field (Theorem 5.9.2). This was sufficient for the proof that each Hilbertian PAC field is ω-free (Theorem 5.10.3).

In this chapter we lay the foundation to the proof of the third major result of this book: Giving a function field E of one variable over an ample field K of cardinality m, each finite split embedding problem over E has m linearly disjoint solution fields (Theorem 11.7.1).

Here we let K ₀ be as in the first paragraph, and consider a finite Galois extension E′ of K ₀(x) (where E′ is not necessarily of the form K(x) with K/K ₀ Galois) acting on a finite group H. We prove that the finite split embedding problem Gal(E′/K ₀(x))⋉H→Gal(E′/K ₀(x)) has a solution field F′. Moreover, if H is generated by finitely many cyclic subgroups G _j , then for each j there is a branch point b _j with G _j as an inertia group.