Algebraic Patching

Abstract

Let E be a field, G a finite group, and {G _i |i∈I} a finite set of subgroups of G with G=〈G _i |i∈I〉. For each i∈I we are given a Galois extension F _i of E with Galois group G _i . We suggest a general method how to ‘patch’ the given F _i ’s into a Galois extension F with Galois group G (Lemma 1.1.7). Our method requires extra fields P _i , all contained in a common field Q and satisfying certain conditions making $\mathcal{E}=(E,F_{i},P_{i},Q;G_{i},G)_{i\in I}$ into ‘patching data’ (Definition 1.1.1). The auxiliary fields P _i in this data substitute, in some sense, analytic fields in rigid patching and fields of formal power series in formal patching.

If in addition to the patching data, E is a Galois extension of a field E ₀ with Galois group Γ and Γ ‘acts properly’ (Definition 1.2.1) on the patching data $\mathcal{E}$, then we construct F above to be a Galois extension of E ₀ with Galois group isomorphic to Γ⋉G (Proposition 1.2.2).