剑桥大学——伽罗华理论.pdf

GaloisTheory Dr P M H Wilson Mich lmas L A T E XedbyAnneHarrison commentsto soc archim notes lists cam ac uk Chapter Introduction Thesenotesarebasedonthecourse GaloisTheory givenbyDr P M H Wil soninCambridgeintheMich lmasTerm Thesetypesetnotesare totallyunconnectedwithDr Wilson Othersetsofnotesareavailablefordi erentcourses Thelistofavailable coursesvariesovertimeandmaybeobtainedfrom http www cam ac uk CambUniv Societies archim notes htm oryoucanemailsoc archim notes lists cam ac uktogetacopyofthe setsyourequire Copyright c TheArchimedeans CambridgeUniversity Allrightsreserved Redistributionanduseofthesenotesinelectronicorprintedform with orwithoutmodi cation arepermittedprovidedthatthefollowingconditions aremet Redistributionsoftheelectronic lesmustretaintheabovecopyright notice thislistofconditionsandthefollowingdisclaimer Redistributionsinprintedformmustreproducetheabovecopyright notice thislistofconditionsandthefollowingdisclaimer 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EVENIFADVISEDOFTHEPOSSIBILITYOF SUCHDAMAGE Chapter RevisionfromGRF FieldExtensions SupposeK L elds recallthatanynon zeroringhomomorphism HM K Lisnecessarilyinjectiveandthen a b a b forb ie isa homomorphismof elds De nitionA eldextensionofKconsistsofa eldLandanon zero eld HM K L RemarkSucha isalsocalledanembeddingofKintoL OfcourseKcan beasub eldofLwith theinclusion InfactweoftenjustidentifyKwith itsimage K L If K La eldextension Lhasthestructure ofaK vectorspace it sanabeliangroupwithKactingviaa a multiplicationinL a K L Thedimensionofthisvectorspaceis calledthedegree L K oftheextension SayLis niteoverKifthevector spaceis nitedimensional ExampleK fp q p p q Qgasub eldofCisa niteextensionof Qofdegree p q p p q p p q LemmaIffK i g i I isanycollectionofsub eldsofa eldLthen T i I K i alsoasub eld ProofEasyexercisefromtheaxioms CHAPTER REVISIONFROMGRF De nitionGivenasub eldk LandS Lanysubset thesub eld generatedbykandS k S fsubfieldsK LjK kandK Sg LemmaThisisasub eldanditisthesmallestsub eldcontainingkand S IfS fx x n g writek x x n fork S Wesaythata eld extension k Lis nitelygenerated f g ifforsomen x x n L stL k x x n Ifmoreovern theextensioniscalledsimple NotationFromnowonweusuallydenotea eldextensionbyk K K kor K j k Givena eldextension k KandasubsetS Kdenotethe eld extensionk k Sbyk S k De nitionGivena eldextensionK k anelementx Kisalgebraic overkif non zeropolynomialf Ks t f x inK otherwisexis calledtranscendental Ifxalgebraicoverk themonicpolynomial f X n a n X n a X a ofsmallestdegreens t f x is calledtheminimalpolynomial Clearlysuchanfisirreducible Remainder Theorem uniqueness De nitionK kisalgebraicifeveryxinKisalgebraicoverk Itiscalled puretranscendentalifnox Kisalgebraicoverkapartfromthosein image of k Theorem Givena eldextension K k x K xalgebraicoverk k x k Whenxisalgebraic k x k isthedegreeofitsminimalpolynomial Proof If k x k n then x x n arelinearlydependentover k polynomialfasclaimedwithf x inK Ifxalgebraicoverkwithminimalpolynomialf thenf x x n a n x n a x a inK Supposeg k x s t g x sincefisirreduciblewehaveh c f f g Euclid sAlgorithm k x s t f g ink x x g x in Kandsog x x x subspaceofKgeneratedbypowersofx FIELDEXTENSIONS Nowk x consistsofallelementsoftheform h x g x forh g k x g x suchelementsformasub eldanditisthesmallestonegeneratedbykand x andsok x spannedasak vectorspaceby x x andhencefrom by x x n Minimalityofn spanningset x x n abasisandhence k x k n Given eldkanirreduciblepolynomialf k x recallthatthequotient ringk x f isa eld Euclid salgorithmyieldsinversesasabove Thereforewehaveasimplealgebraicextensionk k x k x f xde notesimageofX However givenanysimplealgebraicextensionk k x weletfbeminimal polynomialofxoverk Wethenhavea commutation diagram k i k X XinducinganIMof elds evaluation k X f k x k x x Thus uptoIMs anysimplealgebraicextensionofkisoftheformk k X f forf k X irreduciblesoclassifyingsimplealgebraicextensions ofk uptoIM isequivalenttoclassifyingirreduciblemonicpolynomialsin k X Testsforirreducibility seeGRF SupposeRisaUFDandkits eldoffractionseg R Z k Q Gauss sLemmaApolynomialf R X irreducible itisirreduciblein k X EisenstiensIrreducibilitycriterionWithRandkasabove Suppose f a n X n a X a R X andirreducibleallp Rs t pj a n pja i fori n p ja thenfisirreducibleinR X andhencebyGauss s Lemmaink X Ifk K L wehaveatowerof eldextensionsandwe write L j K j k CHAPTER REVISIONFROMGRF Proposition TowerLaw Givensuchatowerof eldextensions L K L K K k usualconventionswith ProofObservethatif L k then K k sinceKasub eldof L and L K sinceanybasisforLoverkisaspanningsetforL K Sow l o g K k m L K n RTP L k mn Letu u m beabasisforLoverK Andv v m beabasisforKoverk ClaimThemnelementsu i v i formabasisforLasavectorspaceoverk easycheckthattheyspanLoverkandarelinearlyindependentoverk Corollory IfKisaf g eldextensionofk sayK k a a m andeacha i algebraicoverk thenK kisa niteextension ProofEacha i algebraicoverk a a i k a a i k a a i i Inductionand K k SplittingFields RecallthatifK kisa eldextensionandf k X wesaythatfsplitsover Kif theimageof finK X splitsintolinearfactors f c X X n c k i kKiscalledasplitting eld orsplittingextension forf ifffailstosplitoveranypropersub eldofK Clearlyequivalenttosaying K k n SplittingFieldsalwaysExist Sinceifgisanyirreduciblefactoroffink x thenk X g k x isan extensionofkforwhichg x x imageofX RemainderTheorem g andhencef splitsofalinearfactor Induction splitting eldKforfwith K k n n degfby Splitting eldsareuniqueuptoisomorphismfollowsfrom SPLITTINGFIELDS Proposition Suppose k k isanisomorphismof eldswith f k X correspondingtog f k X Thenanys f Kforfoverkis isomorphic over toanys f K forgoverk ie commutativediagram K K k k Proof RemarkThuswehavetheexistanceanduniquenessofsplitting eldsfor any nitesetofpolynomials justtakethes f oftheproductofthem with appropriateuseofZorn slemma thisextendstoanysetofpolynomialscf ch whereweprovetheexistanceanduniquenessofalgebraicclosures CHAPTER REVISIONFROMGRF Chapter Separability De nitionAnirreduciblepolynomialf k X iscalledseparableoverk ifithasdistinctzerosinasplitting eldK ief c X X n in K X withc k i distinct Byuniquesness uptoIM ofsplitting elds thiisindependentofanychoices Anarbitrarypolynomialf k X isseparableoverkifallitsirreducible factorsare Iffisnotseparableitiscalledinseparable Todeterminewhetheranirreduciblepolynomialfhasdistinctrootsinas f weintroduceformaldi erentiationofpolynomials D k X k X alinearmapasvectorspacesoverk De nedby D X n nX n n D c c k ClaimD fg fD g gD f ProofUsinglinearity wecanreducethistothecasewherefandgare monomialsinwhichcaseitisatrivialcheck FromnowondenoteD f byf Lemma Apolynomial f k X hasarepeatedrootinasplitting eldi fandf haveacommonfactorofdegree CHAPTER SEPARABILITY Proof Supposefhasarepeatedzero inas f Kief X g inK X Thenf X g X g f f havecommonfactor X inK X f f havecommonfactorink X namely theminimalpolynomialof Supposefhasnorepeatedrootsinas f K Weshowthatf f havenocommonfactorinK X andsoalsoink X Su cienttoprove X jfinK X j f Writingf X g with X j gweobtainf X g gandso X j f Ifnowfisirreducible saysthatfhasrepeatedrootsinas f f Butiff a n X n a X a thenf na n X n a Thereforef ia i inK i Soifdegf n f chark p andpjiwhenevera i fis ofform f b r X pr b r X p r b X p b k X p Soifchark allpolynomialsareseparable Ifchark p anirreduciblepolynomialfinseparable f k X p De nitionGivena eldextensionK kandanelement K wesay that isseparableoverkifitsminimalpolynomialf k X isseparable Theextensioniscalledseparableif separable K Otherwiseitis calledinseparable ExampleLetK F p t eldofrationalfunctionsover nite eldF p with pelements pprime Letk F p t p The eldextensionK kisinseparable theminimalpolyno mialoftoverkisX p t p k X InK X thissplitsas x t p andsois inseparable Lemma Ifwehaveatowerof nite eldextensionsk K Lwith L k separablethenbothL KandK kareseparable ProofGiven L theminimalpolynomialof overKdividesthe minimalpolynomialof overkandsoagainhasdistinctzerosinas f K kseperableobviousfromthede nition Conversetrue butmoredi cultandneedsalittlepreparation Proposition Letk kbea niteextension withf k X themin imalpolynomialfor Givena eldextension k K thenumberof embeddings k Kextending ispreciselythenumberofdistinct rootsof f inK Inparticular atmostn k k suchembeddings withequalityi f splitscompletelyinKandfisseparable ProofThisisessentiallyclear anembeddingk Kextending must send toarootof f andisdeterminedbythisinformation ieif isa rootof f inK thentheringHM k X K g g factorstogivean embedding k k X f Kextending andsending to therefore atmostn degf k k suchembeddings andwehaveequality Leftrightarrow f hasndistinctrootsinK f splitscompletelyin Kandfseparable Theorem SupposeK k a a r isa niteextensionofkandL k any eldextensionforwhichalltheminimalpolynomialsofthea i split ThenumberofembeddingsK Lextendingk Lisatmost K k Ifeacha i isseparableoverK k a a i thenwehaveequality IfthenumberofembeddingsK Lextendingk Lis K k then K kseparable Hence ifeacha i separableoverK k a a i thenK ksep arable by thishappensforinstancewheneacha i separableover k Corollory Ifwehaveatowerk K Lof niteextensionswithL K andK kseparable thensotooisL k Proof Proofof CHAPTER SEPARABILITY Lemma LetG K bea nitesubgroupofthemultiplicativegroup K Knf g Ka eld ThenGiscyclic ProofSeeGRF Theorem PrimativeElementTheorem IfK k a niteextensionodkwith separableoverk Ks t K k Any niteseparableextensionK kissimple Proof TraceandNorm Letk kbea nite eldextension K Multiplicationby de nesalinear map K Kofvectorspacesocerk Thetraceandnormof Tr K k Nm K k arede nedtobethetrace anddeterminantof i e ofanymatrixrepresenting w r t somebasisof Koverk Proposition Supposer K k andf X n a X a minimalpolynomialof overk Ifb i n i a i thenTr K k rb n Nm K k b r Proof CHAPTER SEPARABILITY Chapter AlgebraicClosure De nitionA eldKisalgebraicallyclosedifanyf k X splitsinto linearfactorsoverk Thisisequivalenttotherebeingnonon trivialalgebraic extensionsofK ieanyalgebraicextensionK LisanIM Anextension K kiscalledanalgebraicclosureofkifK kalgebraicandKalgebraically closed Lemma SupposeK kisa eldextension ThesetofelementsofK whicharealgebraicoverkformasub eldLofK IfKalgebraicallyclosed theextensionL kisanalgebraicclosureofk Proof ExampleAlgebraicnumbersinCformasub eld thealgebraicclosureof Q C ExistenceofAlgebraicClosure Asimplemindedapproachtothisleadstosettheoreticproblems theap proachweadoptisperhapsnotthemostnaturalone butavoidsthesprob lemsinacleanway Theorem Analgebraicclosureexistsforany eldk CHAPTER ALGEBRAICCLOSURE Proof Claim Claim UniquenessofAlgebraicClosure Proposition Supposethati k KwithKalgebraicallyclosed For anyalgebraicextensionk L embeddingj L Kextendingi ie L k j i K Proof Corollory Ifi k K i k K aretwoalgebraicclosuresofk then anIM K K s t i i CHAPTER ALGEBRAICCLOSURE Proof RemarkForgeneral eldk theconstructionandproofofuniquenessof algebraicclosure khasinvolvedZorn sLemma andsopreferableingeneral toavoiduseof k NothoweverthatwecanconstructCbybarehands withoutuseofthe axiomofchoice sothisobjectionislessvalidfore g k R Q algebraic number eld itisoftenusefultochoseaparticularzeroofapolynomialin Ce g arealroot Chapter NormalExtensionsandGalois Extensions De nitionAnextensionK kisnormalifeveryirreduciblepolynomial f k X havingarootinKsplitscompletelyoverk oneout allout ExampleQ QisnotnormalsinceX doesn tsplitoveranyreal eld Theorem AnextensionK kisnormaland nite K kisasplitting eldforsomepolynomialoverk Proof Claim CHAPTER NORMALEXTENSIONSANDGALOISEXTENSIONS ProofofClaim NormalClosures Givena niteextensionK k wewriteK k r withf i minimal polynomialof i overkandletL Kbeasplitting eldforF f f r consideredasapolynomialinK X Then L knormal itiscalled thenormalclosureforK k AnyextensionM KforwhichM Knormal mustsplitFandsoforsome sub eldL M L Kisalsoasplitting eldforF andisomorphicoverK toL Kby ThusthenormalclosureofK kmaybecharacterisedasa minimalextensionforwhichL kisnormal anditisuniquuptoIM Note Zorn sLemmanotrequired De nitionGiven eldextensionsK kandL k ak embeddingofKinL isgivenbyacommutativediagram K L k Inthecasewhere K k L kandtheextensionis nite thenK Kisalsosurjective an injectivelinearmapof nitedimensionalvectorspacetoitselfisanIM and hencea eldIMoverk Thesearecalledk automorphisms k AM Much ofGaloisTheoryisconcernedwiththegroupAut K k ofallk AMsof extensionK k Theorem IfK kisa niteextension let K Lbeanormal closureofK kwithK K L Numberoffk embeddingsK Lg K k withequalityi K k separable K knormal anyk embeddingK LhasimageK anyk embeddingK isoftheform forsomek AM ofK Proof Corollory IfK kisa niteextension thenjAut K k j K k with equalityi K kisnormalandseparable Proof Fromnowwewilldealwith eldextensionsk K sub eld wedon t losegeneralitybydoingthisnow asforanyextesnionk K wecanalways identifykwithitsimage CHAPTER NORMALEXTENSIONSANDGALOISEXTENSIONS De nitionIfKa eldandGis nitegroupofAMsofK wedenotethe xedsub eldK G Kwhere K G fx Ks tg x x g Gg Easycheckthisisasub eld Wesaythata niteextensionk KisGaloisifk k G forsome niteGof AMs ClearlyG Aut K k andinfactG Aut K k RemarkBeforewehavetakenthe bottomup approach takingexten sionsofbase elds Heretheapproachis topdown Wewillseethatthese twowaysofdevelopingGaloisTheoryareequivalent Proposition LetGbea nitegroupsofAMsactingona eldK with k K G K Everyelement Khas k k jGj K kseparable K k nitewith K k jGj Proof Theorem Letk Kbea niteextensionof elds Thenthefollowing areequivalent K kgalois kisthe xed eldofAut K k jAut K k j K k K knormalandseparable Moreover ifk K G forsome nitegroupGofAMs wehaveG Aut K k Proof NotationIfk Kgalois weoftenwriteGal K k forAut K k the galoisgroupoftheextension FundamentalTheoremofGaloistheory LetK kbea niteextensionof elds ThegroupG Aut K k has jGj K k by LetF K G k jGj K F IfHasubgroupofG thenthe xed eldL K H isanintermeadiate eldF L Kand Aut K L H Foranyintermeadiate eldF L K letHbethesubgroup Aut K L ofG Aut K k ClaimK LisaGaloisextensionandL K H Proof CHAPTER NORMALEXTENSIONSANDGALOISEXTENSIONS ConclusionTheoperationsH G F K H KandF L K Aut K L Garemutuallyinverse Theorem FundamentalTheoremofGaloisTheory Withnota tionasabove orderreversingbijectionbetweensubgroupsHofGandintermediate eldsF L K whereasubgroupHcorrespondstoits xed eldL K H andanintermediate eldF L KcorrespondstoGal K L G AsubgroupHofGisnormali theextensionK H FisGalois K H Fisnormal IfH G themap G j K HdeterminesagroupHMofGonto Gal K H F withkernelH andhenceGal K H F G H Proof Alreadydone GaloisGroupsofPolynomials Supposenowf k X aseparablepolynomialandK kasplitting eld The galoisgroupoff Gal f Gal K k SupposenowfhasdistinctrootsinK say d soK k d Sinceak AMofKisdeterminedbyitsextensionontheroots i wehavean injectionHM G S d Propertiesoffwillbere ectedintheproperties ifG Lemma Withassumptionsasabove f k X irreducible Gacts transitivelyonrootsoff ie G isatransitivesubgroupofS d Proof Soforlowdegreegaloisgroupsveryrestricted Degree Eitherfreducible G orirreducible G C Degree Eitherfreducible G C orirreducible G S C QuestionForf k X irreducibleandseparableofdegreed whenisthe imageofgaloisgroupinA d De nitionThediscriminantDofapolynomialf k X withdistinct rootsinas f eg firreducibleandseparable isde nedasfollows Let d berootsoffinas fKandset Q i j i j Thediscrim inant D Y i j i j d d is xedbyalltheelementsofG Gal K k andhenceisanelementofk Assumingfirreducibleandseparablewehave and assumingchar G A d xedunderG sinceforanyodd K Dasquareink CHAPTER NORMALEXTENSIONSANDGALOISEXTENSIONS Example Cubicsf X bX cX dandchar firreducible separable ThegaloisgroupGisA C i D f asquareinkandS otherwise To calculateD f setg X f X b offormX pX q Since aroot off b arootofgdeduce f g D f D g Lemma Iffirreducibleseparablepolynomialink X L kasplitting eld LarootoffandK k Lthen D f d d Nm K L f Proof Example Whenk Q wecanconsiderthes f ofpolynomialf Q X asasub eld ofC thismaybeuseful e g iff Q X irreducibleofdegreedwithpreciselytwoimaginaryroots thegaloisgroupcontainsatransposition complexconjugationisanelement ofGal f andswitchestwoimaginaryroots ElementarygrouptheoryshowsthatifG S p pprime istransitiveand containsatransposition thenitcontainsalltranspositionsandhenceG S p Manycubicsf Q X havegaloisgroupS forthisreason Itisahelpto knowwhatthetransitivesubgroupsofS n are ifwearetryingtocalculate thegaloisgroupofapolynomial Thefollowingclassi cationforn is leftasanextendedexerciseingrouptheory Proposition ThetransitivesubgroupsofS areS A D andV C C thegroupfid g ThetransitivesubgroupsofS areS A G D C whereG isgenerated bya cycleanda cycle RemarkAllthesepossibilitiesoccur Chapter GaloisGroupsofFiniteFields RecallIfFa eldwithjFj q thenq p r wherecharF p De nitionGivensucha niteF F p AM F Fgiven x x p calledthefrobeniusautomorphism since x y p x p y p andx p x isaninfective eld HMF F whichissurjectivesincejFjis nite andhenceanAM Now observex p x x F p theprimesub eld Moreover sincea q F wehavea q a a F i e everyelement ofFisarootofX q X ButX q Xhasatmostqroots sotheseareall theroots thereforeFisthes f ofX q XoverF p andassuchisuniqueup toIM ExistanceIfq p r letFbethes f ofX q XoverF p wehavethe frobeniusAm F Fandwecantakethe xed eldF ofh r i Observethat r x x xisarootofX q XandsoF Fi