Split Embedding Problems over Ample Fields

Abstract

We generalize Theorem 5.9.2 and prove that if K 0 is an ample field, then not only constant finite split embedding problems over K 0(x) are solvable but every finite split embedding problem Gal(E/K 0(x))⋉H→Gal(E/K 0(x)) has as many linearly disjoint solution fields F α , with α<card(K) (Proposition 8.6.3). Moreover, let K be the algebraic closure of K 0 in E. Then each K-rational place φ of E unramified over K 0(x) with φ(x)∈K 0∪{∞} extends to a K-rational place of F α unramified over K 0(x) (Lemma 8.6.1). The construction of the solutions for general finite split embedding problems over K 0(x) in the case where K 0 is an ample field relies on Proposition 7.3.1, where K 0 is assumed to be complete under an ultrametric absolute value. For an arbitrary ample field K 0, we first lift the embedding problem to one over K 0((t))(x), and apply Proposition 7.3.1 to solve it with additional information on the branch points (in particular they should be algebraically independent over K 0) and on their inertia groups. Then, we use Bertini-Noether as in Lemma 5.9.1 to reduce that solution to one of the original problem. In order to achieve many linearly disjoint solutions the reduction has to keep track of the branch points and their inertia groups. This can be done once the reduction is normal in the sense of Section 8.1.