Galois theory, coverings, and Riemann surfaces PDF
Preview Galois theory, coverings, and Riemann surfaces
Galois Theory, Coverings, and Riemann Surfaces Askold Khovanskii Galois Theory, Coverings, and Riemann Surfaces AskoldKhovanskii Dept.Mathematics UniversityofToronto Toronto,Ontario,Canada ISBN978-3-642-38840-8 ISBN978-3-642-38841-5(eBook) DOI10.1007/978-3-642-38841-5 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013949677 MathematicsSubjectClassification(2010): 55-02,12F10,30F10 ©Springer-VerlagBerlinHeidelberg2013 TranslationofRussianeditionentitled“TeoriyaGalua,NakrytiyaiRimanovyPoverkhnosti”,published byMCCME,Moscow,Russia,2006 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface The main goal of this book is an exposition of Galois theory and its applications to the questions of solvability of algebraic equations in explicit form. Apart from the classical problem on solvability of an algebraic equation by radicals, we also considerotherproblemsofthistype,forinstance,thequestionofsolvabilityofan equationbyradicalsandbysolvingauxiliaryequationsofdegreeatmostk. There exists a surprising analogy between the fundamental theorem of Galois theory and classification of coverings over a topological space. A description of this analogy is the second goal of the present book. We consider several classifi- cations of coverings closely related to each other. At the same time, we stress a formal analogy between the results thus obtained and the fundamental theorem of Galoistheory.Apartfromcoverings,weconsiderfiniteramifiedcoveringsoverRie- mannsurfaces(i.e.,overone-dimensionalcomplexmanifolds).Ramifiedcoverings are slightly more complicated than unramified finite coverings, but both types of coveringsareclassifiedinthesameway. Thethirdgoalofthebookisageometricdescriptionoffinitealgebraicextensions ofthefieldofmeromorphicfunctionsonaRiemannsurface.Forsuchsurfaces,the geometryoframifiedcoveringsandGaloistheoryarenotonlyanalogousbutinfact verycloselyrelatedtoeachother.Thisrelationshipisusefulinbothdirections.On theonehand,GaloistheoryandRiemann’sexistencetheoremallowonetodescribe thefieldoffunctionsonaramifiedcoveringoveraRiemannsurfaceasafinitealge- braicextensionofthefieldofmeromorphicfunctionsontheRiemannsurface.On the other hand, the geometry of ramified coverings together with Riemann’s exis- tence theorem allows one to give a transparent description of algebraic extensions ofthefieldofmeromorphicfunctionsoveraRiemannsurface. Thebookisorganizedasfollows.ThefirstchapterisdevotedtoGaloistheory.It isabsolutelyindependentoftheotherchapters.Itcanbereadseparately. Thesecondchapterisdevotedtocoveringsovertopologicalspacesandtoram- ifiedcoveringsoverRiemannsurfaces.Itisalmostindependentofthefirstchapter. Inthesecondchapter,westressaformalanalogybetweentheclassificationofcov- erings and the fundamental theorem of Galois theory. This is the only connection v vi Preface betweenthechapters—toreadthesecondchapter,itisenoughtoknowtheformu- lationofthefundamentaltheoremofGaloistheory. ThethirdchapterreliesonGaloistheoryaswellasontheclassificationofram- ified coverings over Riemann surfaces. Nevertheless, it can also be read indepen- dentlyifthereaderacceptswithoutproofthenecessaryresultsofthefirsttwochap- ters. Thenumberingoftheorems,propositions,lemmasetc.ineverychapterissepa- rate,butformulasarenumberedconsistentlythroughthewholebook. Thebookisaddressedtomathematiciansandtoundergraduateandgraduatestu- dents majoring in mathematics. Some results of the book (for instance, necessary conditions of various forms of solvability of complicated algebraic equations via solutions of simpler algebraic equations, a description of an analogy between the theoryofcoveringsandGaloistheory)mightbeofinteresttoexperts. Contents 1 GaloisTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 ActionofaSolvableGroupandRepresentabilitybyRadicals . . 2 1.1.1 ASufficientConditionforSolvabilitybyRadicals . . . . 3 1.1.2 ThePermutationGroupoftheVariablesandEquations ofDegree2to4 . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 LagrangePolynomialsandCommutativeMatrixGroups . 5 1.1.4 SolvingEquationsofDegree2to4byRadicals . . . . . 8 1.2 FixedPointsunderanActionofaFiniteGroupandItsSubgroups 11 1.3 FieldAutomorphismsandRelationsBetweenElementsinaField 14 1.3.1 SeparableEquations . . . . . . . . . . . . . . . . . . . . 14 1.3.2 AlgebraicityovertheInvariantSubfield . . . . . . . . . . 15 1.3.3 SubalgebraContainingtheCoefficientsoftheLagrange Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.4 RepresentabilityofOneElementThroughAnother ElementovertheInvariantField. . . . . . . . . . . . . . 17 1.4 Actionofak-SolvableGroupandRepresentabilitybyk-Radicals 18 1.5 GaloisEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.6 AutomorphismsConnectedwithaGaloisEquation . . . . . . . . 21 1.7 TheFundamentalTheoremofGaloisTheory . . . . . . . . . . . 22 1.7.1 GaloisExtensions . . . . . . . . . . . . . . . . . . . . . 22 1.7.2 GaloisGroups . . . . . . . . . . . . . . . . . . . . . . . 23 1.7.3 TheFundamentalTheorem . . . . . . . . . . . . . . . . 24 1.7.4 PropertiesoftheGaloisCorrespondence . . . . . . . . . 24 1.7.5 ChangeoftheCoefficientField . . . . . . . . . . . . . . 25 1.8 ACriterionforSolvabilityofEquationsbyRadicals . . . . . . . 27 1.8.1 RootsofUnity . . . . . . . . . . . . . . . . . . . . . . . 27 1.8.2 TheEquationxn=a . . . . . . . . . . . . . . . . . . . 28 1.8.3 SolvabilitybyRadicals . . . . . . . . . . . . . . . . . . 29 1.9 ACriterionforSolvabilityofEquationsbyk-Radicals . . . . . . 30 1.9.1 Propertiesofk-SolvableGroups. . . . . . . . . . . . . . 30 vii viii Contents 1.9.2 Solvabilitybyk-Radicals . . . . . . . . . . . . . . . . . 32 1.9.3 UnsolvabilityofaGenericDegree-(k+1>4)Equation ink-Radicals . . . . . . . . . . . . . . . . . . . . . . . . 33 1.10 UnsolvabilityofComplicatedEquationsbySolvingSimpler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.10.1 ANecessaryConditionforSolvability . . . . . . . . . . 35 1.10.2 ClassesofFiniteGroups . . . . . . . . . . . . . . . . . . 36 1.11 FiniteFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2 Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.1 CoveringsoverTopologicalSpaces . . . . . . . . . . . . . . . . 42 2.1.1 CoveringsandCoveringHomotopy . . . . . . . . . . . . 42 2.1.2 ClassificationofCoveringswithMarkedPoints . . . . . 43 2.1.3 CoveringswithMarkedPointsandSubgroups oftheFundamentalGroup . . . . . . . . . . . . . . . . . 45 2.1.4 CoveringsandGaloisTheory . . . . . . . . . . . . . . . 48 2.2 CompletionofFiniteCoveringsoverPuncturedRiemannSurfaces 52 2.2.1 FillingHolesandPuiseuxExpansions . . . . . . . . . . 52 2.2.2 Analytic-TypeMapsandtheRealOperationofFilling Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2.3 FiniteRamifiedCoveringswithaFixedRamificationSet 57 2.2.4 RiemannSurfaceofanAlgebraicEquationovertheField ofMeromorphicFunctions . . . . . . . . . . . . . . . . 62 3 RamifiedCoveringsandGaloisTheory . . . . . . . . . . . . . . . . 65 3.1 FiniteRamifiedCoveringsandAlgebraicExtensionsofFields ofMeromorphicFunctions . . . . . . . . . . . . . . . . . . . . 66 3.1.1 TheFieldP (O)ofGermsatthePointa∈X a ofAlgebraicFunctionswithRamificationoverO . . . . 66 3.1.2 GaloisTheoryfortheActionoftheFundamentalGroup ontheFieldP (O) . . . . . . . . . . . . . . . . . . . . 68 a 3.1.3 FieldofFunctionsonaRamifiedCovering . . . . . . . . 71 3.2 GeometryofGaloisTheoryforExtensionsofaField ofMeromorphicFunctions . . . . . . . . . . . . . . . . . . . . 72 3.2.1 GaloisExtensionsoftheFieldK(X) . . . . . . . . . . . 72 3.2.2 AlgebraicExtensionsoftheFieldofGerms ofMeromorphicFunctions . . . . . . . . . . . . . . . . 73 3.2.3 AlgebraicExtensionsoftheFieldofRationalFunctions . 74 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Chapter 1 Galois Theory In this first chapter, we give an exposition of Galois theory and its applications to thequestionsofsolvabilityofalgebraicequationsinexplicitform. InSects.1.1–1.4,weconsiderafieldP onwhichafinitegroupGactsbyfield automorphisms. Elements of the field P fixed under the action of G form a sub- fieldK⊆P,whichiscalledtheinvariantsubfield. In Sect. 1.1, we show that if the group G is solvable, then the elements of the fieldP arerepresentablebyradicalsthroughtheelementsoftheinvariantfieldK. (Here,anadditionalassumptionisneededthatthefieldK containsallrootsofunity of order equal to the cardinality of G.) If P is the field of rational functions of n variables, then G is the symmetric group acting by permutations of the variables, andK isthesubfieldofsymmetricfunctionsofnvariables.Thisresultprovidesan explanationforthefactthatalgebraicequationsofdegrees2to4inonevariableare solvablebyradicals. InSect.1.2,weshowthatforeverysubgroupG ofthegroupG,thereexistsan 0 elementx∈P whosestabilizerisequaltoG .TheresultsofSects.1.1and1.2are 0 basedonsimpleconsiderationsfromgrouptheory;theyuseanexplicitformulafor theLagrangeinterpolatingpolynomial. In Sect. 1.3, we show that every element of the field P is algebraic over the fieldK.Weprovethatifthestabilizerofapointz∈P containsthestabilizerofa pointy∈P,thenzisthevalueatyofsomepolynomialoverthefieldK.Thisproof isalsobasedonthestudyoftheLagrangeinterpolatingpolynomial(seeSect.1.3.3). In Sect. 1.4, we introduce the class of k-solvable groups. We show that if a group G is k-solvable, then the elements of the field P are representable in k- radicals (i.e., can be obtained by taking radicals and solving auxiliary algebraic equations of degree k or less) through the elements of the field K. (Here, we also needtoassumeadditionallythatthefieldK containsallrootsofunityoforderequal tothecardinalityofG.) Consider now a different situation. Suppose that a field P is obtained from a field K by adjoining all roots of a polynomial equation over K with no multiple roots. In this case, there exists a finite group G of automorphisms of the field P whoseinvariantfieldcoincideswithK.ToconstructthegroupG,theinitialequa- A.Khovanskii,GaloisTheory,Coverings,andRiemannSurfaces, 1 DOI10.1007/978-3-642-38841-5_1,©Springer-VerlagBerlinHeidelberg2013 2 1 GaloisTheory tionneedstobereplacedwithanequivalentGaloisequation,i.e.,withanequation each of whose roots can be expressed through any other root (see Sect. 1.5). This groupGofautomorphismsisconstructedinSect.1.6. ThusSects.1.2,1.3,1.5,and1.6containproofsofthecentraltheoremsofGalois theory.InSect.1.7,wesummarize,thenstateandprove,theFundamentaltheorem ofGaloistheory. AnalgebraicequationoverafieldissolvablebyradicalsifandonlyifitsGalois groupissolvable(Sect.1.8),anditissolvablebyk-radicalsifandonlyifitsGalois group is k-solvable (Sect. 1.9). In Sect. 1.10, we discuss the question of solvabil- ityofalgebraicequationswithhighercomplexitybysolvingequationswithlower complexity.WegiveanecessaryconditionforsuchsolvabilityintermsoftheGalois groupoftheequation. InSect.1.11,weclassifyfinitefields.Wecheckthatthefundamentaltheoremof Galoistheoryholdsforfinitefields(theproofofthefundamentaltheoremgivenin Sect.1.7doesnotgothroughforfinitefields). Inthischapter,amajorfocusisontheapplicationsofGaloistheorytoproblems of solvability of algebraic equations in explicit form. However, the exposition of Galois theory does not refer to these applications. The fundamental principles of Galois theory are covered in Sects. 1.2, 1.3, 1.5–1.7. These sections can be read independentlyoftherestofthechapter. Arecipeforsolvingalgebraicequationsbyradicals(includingsolutionsofgen- eralequationsofdegree2to4)isgiveninSect.1.1,andisindependentoftherest of the text. The classification of finite fields given in Sect. 1.11 is also practically independentoftherestofthetext.Thesesectionscanbereadindependentlyofthe restofthechapter. 1.1 ActionofaSolvable Groupand Representabilityby Radicals Inthissection,weprovethatifafinitesolvablegroupGactsonafieldP byfield automorphisms, then (under certain additional assumptions on the field P), all el- ements of P can be expressed through the elements of the invariant field K by radicalsandarithmeticoperations. A construction of a representation by radicals is based on linear algebra (see Sect. 1.1.1). In Sect. 1.1.2, we use this result to prove solvability of equations of lowdegrees.Toobtainexplicitsolutions,thelinearalgebraconstructionneedstobe done explicitly. In Sect. 1.1.3, we introduce the technique of Lagrange resolvents, whichallowsustoperformanexplicitdiagonalizationofanAbelianlineargroup. InSect.1.1.4,weexplain,howLagrangeresolventscanhelptowritedownexplicit formulaswithradicalsforthesolutionsofequationsofdegree2to4. The results of this section are applicable in the general situation considered in Galois theory. If a field P is obtained from the field K by adjoining all roots of an algebraic equation without multiple roots, then there exists a group G of auto- morphismsofthefieldP whoseinvariantfieldisthefieldK (seeSect.1.7.1).This
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