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).
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© 2011 Springer-Verlag Berlin Heidelberg
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Jarden, M. (2011). Algebraic Patching. In: Algebraic Patching. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15128-6_1
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DOI: https://doi.org/10.1007/978-3-642-15128-6_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15127-9
Online ISBN: 978-3-642-15128-6
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