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Patching and Galois theory - Penn Math

Patchingand GaloistheoryDavidHarbater Mathematics,University of PennsylvaniaAbstract:Galoistheoryover (x) is well-understood asa consequenceof Riemann sExistenceTheorem,which classifiesthealgebraicbranchedcoversof of thattheoremusesanalyticandtopologicalmet hods,includingtheabilitytoconstructcover slocallyandtopatch studytheGaloisextensionsofk(x) forotherfieldsk, onewouldlike to have ananalogof Riemann sExistenceTheoremforcurves overk. Such a resultremainsoutof reach, butpartialresultsin thisdirectioncanbe provenusingpatchingmethodsthatareanalogo ustocomplexpatching,andwhich applyin method is formalpatching,in which formalcompletionsof schemesplay theroleof is rigidpatching,in which yieldtherealizationof arbitraryfinitegroupsas Galoisgroupsoverk(x) forvariousclassesof fieldsk, as well as suc

Patching and Galois theory David Harbater Dept. of Mathematics, University of Pennsylvania Abstract: Galois theory over (x) is well-understood as a consequence of Riemann’s

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