; TeX output 2001.06.04:1747o$VK`y cmdunh10FORMSٚOFHOPFALGEBRASANDGALOISٚTHEORYK`y cmr10BODOUUP*AREIGIS%BXQ cmr12ThetheoryofHopfalgebrasiscloselyconnectedwithvXariousap-6plications,cin particulartoalgebraicandformalgroups.oAlthoughthe6 rst'oSccurenceofHopfalgebraswrasinalgebraictopologyV,?theyarenorw6foundinareasasremoteascomrbinatoricsandanalysis.Theirstruc-6ture0hasbSeenstudiedingreatdetailandmanryoftheirpropertiesare6wrellunderstoSod.ϚWVeareinrterestedinasystematictreatmentofHopf6algebraswiththetecrhniquesofformsanddescent.BThep rstthreeparagraphsofthispapSergivreasurveyofthepresent6statec|ofthetheoryofformsofHopfalgebrasandofHopfGaloistheory6espSeciallyHforseparableextensions.Itincludesmanryillustratingexam-6plesesomeofwhicrhcannotbSefoundindetailintheliterature. Thelast6trwomBparagraphsaredevrotedtosomeneworpartialresultsonthesame6 eld. isabilinearmapwithinducedmap/^g (:lB R5C68/!JDonjthetensorproSduct,thenwrecande neP2gn9(a(Bd) ;a(C) ):=7K|^6g<4(fG(a))andthrususethe"compSonents"a(Bd)r_anda(C)Casiftheywere6wrell-de ned'ordinaryelements,NwhichcanbSeusedasargumentsin6bilinearmaps.BSimilartotheHopfalgebrakg[x;x21 \|]eacrhcommutative(asanalge-6bra)HopfalgebraHrepresenrtsafunctor8卍x}j~zHk:#kg-QAlg I!*GdBrp W|;x׹~H$)(A)UR:=kg-Algfl(HF:;A)6wherethemrultiplicationonx~H~isgivenbythecommutativediagram@uxY~MH@(A)x~H T>(A)x~H(A):2fd$ά-έkg-QAlgfl(H HF:;A)`kg-QAlgfl(HF:;A)832fd ά- ҟǠ@fe?Ǡ?ҟǠ@fe 1Ǡ?BहSothegroup-ringkgZhasbSeenseentobeaHopfalgebrawiththe6diagonal(gn9){=g' goforevreryelementgointhegroupZ.{~Thisholds6not onlyforthegroupZ. JEvrerygroup-ringkgGisaHopfalgebra6withthesamecomrultiplication,Revenfornon-commrutativegroupsG.6The2non-commrutativegrouprings,}however,do2notanrymorerepresent6groupovXaluedfunctorsonkg-QAlgfl.ȪTheyarespSecialinstancesofformal6groups.BAnotherconcreteexampleofagroupvXaluedfunctoris9C1: uzkg-QAlg I!*GdBrp W|;6हthecirclegroup,ebde nedbryCܞ(A)Y:=f(a;b)2AxAja228+b22  =Y1g.6Then7groupstructureisgivrenby(a;b)(c;d):=(acbd;ad+bc).6The%represenrtingHopfalgebraisthe"trigonometricalgebra"H=6kg[c;s]=(c22j+s221).8Thediagonalisde nedbry9ʍT{(c)Q6=tc cs s;)(s)Q6=tc s+s c:ԍBहThewmostinrterestingobservXationisthis.LetAbSeacommutativekg-6algebraIwith2inrvertibleIandconrtainingiUR=p UTljz5S m91.ThenItheassignmentCor)U@(A)UR3a7!qō ]ݹ1 ]ݟ[z ΍2q (a+a 1 \|);ō4D131[z # ΍2iH(aa 1)qT2Cܞ(A)"6de nesafunctorialisomorphismofgroups.oIf221 \|;i2kthenUand6CareRisomorphicgroupvXaluedfunctors,hencetheyharveRisomorphic6represenrtingHopfalgebras9:0kg[x;x 1 \|]PUR԰n9=k[c;s]=(c 2j+s 21):o$64yBODOTP:AREIGISV6हIfyi=UR2k thenthetrwoygroupvXaluedfunctorsarenotisomorphic,xneither6aretheirrepresenrtingHopfalgebraskg[x;x21 \|]andk[c;s]=(c22X+s221).BIf!ykisa eldofcrharacterictic6=f^2andi˵=2̽kg,o-thenUb]andCare6non-isomorphicmbuttheyinduceisomorphicfunctorsU@jk6(i)andCܞjk6(i)if6restricted^tothekg(i)-algebras. +LetK1=URk(i)andletAbSeaKܞ-algebra.6Thenwrehave+nʍGKܞ-FAlgfl(KF kg[x;x21 \|];A)P͋԰ͤӹ=6kg-QAlgfl(kg[x;x21 \|];A)P͋԰ͤӹ=6U@jK;¹(A)P͋԰ͤӹ=6CܞjK;¹(A)P͋԰ͤӹ=6kg-QAlgfl(kg[c;s]=(c22j+s221);A)P͋԰ͤӹ=6Kܞ-FAlgfl(KF kg[c;s]=(c22j+s221);A)/y6hencekK D^kg[x;x21 \|]Pՙ԰=K kg[c;s]=(c22b+s221)kasKܞ-Hopfalgebras,6wherePthetensorproSductisalwraysPtakenoverthebaseringkg.ZObserve6thatacancellationpropSertrycannotbeexpectedinthiscase.BInQparticular,pGtheQ-HopfalgebrasQ[x;x21 \|]andQ[c;s]=(c2226+r2s221)6and theR-HopfalgebrasR[x;x21 \|]andR[c;s]=(c22.޹+ns221) arenot6isomorphic,:but*theC-HopfalgebrasC[x;x21 \|]P ԰=mC[c;s]=(c22)+%s221)6are.8Thisisanexampleforthenextde nition.{66De nition1.2.ǹLetGandG20]3bSegroupvXaluedfunctorsonkg-QAlgfl.QLet6KOҹbSes4afaithfully atcommrutatives4kg-algebra. ҅Iftherestrictionsto6Kܞ-FAlg areisomorphicgroupvXaluedfunctors:XGjKP ԰ =;G209jK;¹,thenGand6G20arecalledKܞ-formsofeacrhotherasgroups.BLetHandHV20Z6bSeHopfalgebrasorverthecommrutativeringkg.TLet6KbSe+afaithfully atcommrutative+kg-algebra.CIfK) ֋HPa԰H=K HV20歹as6Kܞ-Hopf@^algebras,UthenH-andHV20arecalledK-formsofeacrhotheras6HopfalgebrasBWVeڕsarythatGandG20ιresp.HandHV20$areformsofeachotherif6thereexistsafaithfully atkg-algebraKcsucrhthattheyareKܞ-formsof6eacrhother.{6BSo forGandG20ܹtobSeKܞ-formsofeacrhotherweneedanisomorphism6ofsetvXaluedfunctors h:kGjK  !G209jK &jsucrhthatBDMGjK jGjKG209jK jG20jK™4:2fdJά-ͯ- 7 GjK$fG209jKĞ32fdoˀά-ƍ 2Ǡ@fedǠ?/2Ǡ@fe/dǠ?U%Bहcommrutes.BThereTmarybSemanydi erentHopfalgebrasHV20whichareformsforH6हwithZ3respSecttosomefaithfully atextensionKܞ.Inparticularthericrh-6nessV3ofHopfalgebrasorverV3QshouldbSehigherthanorverV3C.{Granted6theremarybSeHopfalgebrasde nedoverC,whichdonotcomeabSout)o$pIF9ORMS!OFHOPFALGEBRASANDGALOISTHEOR:Y45V6हbryQabaseringextensionfromQ,kbute.g.ndsemisimplecoScommutative6HopfmalgebrasorvermCarealwraysmde nedorvermQ.CThisisaconsequence6ofBiamoregeneralstructuretheoremofMilnor,XZMoSoreandCartieron6coScommrutativeHHopfalgebrasorverHalgebraiclyclosed elds.'kOurinrter-6ests4areinthisricrhnessofHopfalgebrasover"small" elds.Onecan6shorw!7forexamplethatoverthe eldRofrealsthecirclefunctorCչis6theonlynon-trivialformoftheunitsfunctorU@.BThere`isadescriptionofKܞ-formsforquitegeneralalgebraicstruc-6turesgivrenbythetheoryoffaithfully atdescent.SWVeapplyittothe6case)NofHopfalgebras.LetHbSeaHopfalgebraorver)Nkg.Thegroupof6automorphismsMofthisHopfalgebrawillbSedenotedbrykg-QH˼opfe^-PAut&7(HV).6After{abaseringextensionbryko!+NKwegetaHopfalgebraKB f6Hѹover6K$ZwithGgroupofautomorphismsKܞ-FH˼opfe^-PAut&7(Kƥ HV).PEvrerychange6of4thebaseringextension,Ei.e.evreryhomomorphismofcommutativekg-6algebras[zK1K{!LinducesagrouphomomorphismKܞ-FH˼opfe^-PAut&7(Kb 3HV)68/!HOL-H˼opfe^-PAut&7(L2H HV).ѭThruswehaveafunctorAutչ(HV)UT:kg-QAlg I!*GdBrp6हde nedJbryAut<(HV)(Kܞ):=K-FH˼opfe^-PAut&7(K ,HV).YEvreryJgroupvXalued6functor=onthecategorykg-QAlg kofcommrutative=kg-algebrashasanasso-6ciated,AmitsurcohomologyHV2n(K5=kg;AutŹ(HV)).Itisnotnecessaryto6knorw9theprecisede nitionofthesecohomologygroupstoapplythe6follorwingtheorem.6Theorem$1.3.INLffet(H~beakg-Hopfalgebra.CThenthereisabijektion6bffetweenthesetof(isomorphismclassesof=)Kܞ-formsofH=andthe6A2mitsur35cffohomologygroupHV21Z(K5=kg;AutŹ(HV)).BहProSofsofthismarybefoundinvXariousformsin[G],[H],or[KrO].6Actually]fthistheoremholdsingreatergeneralitryandtheproSofisquite6tecrhnicalandinvolved.BIn viewofthistheoremthemainproblemofcalculatingformsisto6determine_thesetofHopfalgebraautomorphismsofaHopfalgebra.6InNOfactwredonothavetocalculatethecohomologygroup,g9sincebya6trwofoldY{applicationofthistheorem{goingfromcertainformstothe6cohomologygroupandthenfromthesamecohomologygroupbacrkto6some?otherforms{wrewilleliminatetheexplicitcomputationofthe6cohomologyV.BInthecaseofgroup-ringskgGof nitelygeneratedgroupsGthe6automorphismgroupkg-QH˼opfe^-PAut&7(kgG)canbSecalculated,inparticular6forcyclicgroupsCn ofordern.nWVeassumethattheautomorphism6groupmF3ofGis nite./Thenonecanshorwthatkg-QH˼opfe^-PAut&7(kgG)is6isomorphic@jtotheautomorphismgroupG.al 3- Aut!(E2Fykf)ofthetrivialFƹ-6GaloisjextensionE2Fyk -ofkg.ThisGaloisextensioncanbSedescribedbry6theuringE2Fyk =UR(kgFƹ)2,LthedualspaceofthegroupringkFƹ,LonwhicrhF:o$66yBODOTP:AREIGISV6हactsbryautomorphismsinsuchawayV,Mthattheringextension(kgFƹ)2=k6हisg[anFƹ-Galoisextensioninthesenseof[CHR].Actuallythisleadstoa6functorial#iisomorphismAut0(kgG)PUR԰n9=G.al 3- Aut$(E2Ff),KCsothattheAmitsur6cohomologygroupsofthesetrwogroupfunctorsalsocoincide.BWVepformrulateoneofthemostinterestingconsequencesofthesecon-6siderations.h6TheoremƝ1.4.)[HP]"9Lffet8kbeacommutativeringwith2notazero6divisoroinkandPicꠟ(2)!G(kg)N=0,theotwo-torsionofthePicffardogroup.6ThenMRda)\$the35HopfalgebrffaformsofkgZareoHB=URkg[c;s]=(s 2jascbc 2+u);RԍLHb)\$the35HopfalgebrffaformsofkgC39areLHB=URkg[c;s]=(s 2jascbc 2+u;(c+1)(c2);(c+1)(sa));Mc)\$the35HopfalgebrffaformsofkgC49are{HB=URkg[c;s]=(s 2jascbc 2+u;c(ac2s));LHd)\$the35HopfalgebrffaformsofkgC69are6HB=URkg[c;s]=(s 2ascbc 2+u;(c2)(c1)(c+1)(c+2);(c1)(c+1)(sc2a)):BInFallcffasesa;b;uxG2k2satisfyFa22x+4b=uanduisaunitinkg. The6Hopf35algebrffastructureinallcasesisde nedby&(ʍ[ |(c)} 7=URu21 \|((a22j+2b)c ca(c s+s c)+2s s);Z*(s)} 7=URu21 \|(abc c+2b(c s+s c)+as s);`(c)} 7=UR2;(s)=a;n9(c)=c;n9(s)=acs:BहWVe hgivreanindicationofthewayhowthisresultisobtained. Inall6cases/ofthetheoremthegroupFNisthecyclicgroupwithtrwo/elements.6TheRtheoryofC2-Galoisextensions(=quadraticGaloisextensions)is6wrellBknown.ActuallyeveryquadraticGaloisextensionofkisaformof6the}trivialquadraticGaloisextension(kgC2)2PV԰.==k kTofkaswillbSeseen6bSelorw.0SincetheautomorphismgroupsAut߹(kgG)P԰ =9KG.al 3- Aut$(kOk)6coincide,$bthep rstAmitsurcohomologygroupsdescribingtheforms6coincide,_toSo.mSowMthereisabijectivrecorrespondencebetrweenwMtheforms6ofdthegroup-ringsinthetheoremandthequadraticGaloisextensionsof6k+[seeiThm.,!4.1].ThiscorrespSondencewrasusedtoexplicitlycalculate6theformsgivreninthetheorem.Keo$pIF9ORMS!OFHOPFALGEBRASANDGALOISTHEOR:Y47V92.8BHopfGaloisextensionsBहAxdi erenrtyclassof"forms"isobtainedifoneconsidersthefollowing6cancellationproblem.6De nition&2.1.coLet@GUT:kg-QAlg I!*GdBrp ubSeagroupvXaluedfunctor.Then6themrultiplicationofGonitselfG̿G!nGmakesGaG-setvXalued6functor.1Herewrede nethefunctorG~Gعby(G~G)(Kܞ)UR:=G(K)~6G(Kܞ),sothatthemrultipicationofeachgroupG(Kܞ)de nesafunctorial6homomorphismbSG"G!:![G,>brie ythemrultiplicationonGandthe6G-setstructureisde ned"compSonenrtwise".BLetqQXF׹: _kg-QAlg I!*Svet RbSeanotherfunctorwhicrhisalsoaG-setvXalued6functor4bryX)ܦGM!'X.'LetKbSeafaithfully atcommutativering6extension–ofkg.0IftherestrictionsGjK XandXjKtoKܞ-FAlg )areisomorphic6as$GjK;¹-setvXaluedfunctors,3thenGandXarecalledKܞ-formsofeacrh6otherasG-setvXaluedfunctors.BSo&forGandX\tobSeKܞ-formsofeacrhotherweneedanisomorphism6ofsetvXaluedfunctors h:kGjK  !XjK &jsucrhthatExb]GjK jGjKXjK jGjK$D:2fdά-ͳN 71"GjK]XjKhԞ32fdAά-ƍe ZBǠ@fetǠ?LBǠ@fetǠ?ۍBहcommrutes.BAHopfalgebraicdescriptionofthisissomewhatmorecomplicated.6ThenotionofaG-setandofformsofaG-settranslatedtotherepre-6senrting~objectsoftherepresentablefunctorsGandXpgivesthefollow-6ingde nition.6De nition: 2.2.RLetHV2 9bSeacommrutativeHopfalgebraandAbea6commrutativerxalgebra.AiscalledanHV2Z-cffomoduleļalgebrarxifthereisan6algebramapUT:AURn!1A HV2 sucrhthatthediagramsADԠK8AԠ aA HV2W~&:2fd?-ά-t|sf<.aA HV2AA HV2X HV2f&32fd!ά-kTi!n 1OlğǠ@feOǠ?uGEV^ğǠ@feǠ?ԐDv1 d޹andԠ#AԠrA HV20 :2fd?-ά-t|K#Aw"A k0 32fdCά-%M4' Ǡ@fe(/RǠ?*.id Ǡ@fe!RǠ?ꬾ1 u$Bहcommrute.BLet5UHV2 ⯹bSeacommrutative5UHopfalgebraandAbeacommrutative6HV2Z-comoSduleDalgebra. bLetKbeafaithfully atcommrutativeDring6extensionYofkg.IfK_V HV2P ԰ =UK AYŹasK_V HV2Z-comoSdulealgebras,vthen6A꨹iscalledaKܞ-formofHV2Z.X\o$68yBODOTP:AREIGISVBहClosely2+connectedwithKܞ-formsofG-setvXaluedfunctorsistheno-6tionofaprincipalhomogeneousspace.,6De nition2.3.If*GisagroupandX]isaset,zthenaG-setXis6calledWhomoffgeneous,1ifforeacrhpairx;y/2XHthereexistsag2G6हsucrh#thatxg=cyn9.RRA!G-setX䦹isaprincipffal;homogeneous#G-setifXis6homogeneousandxgË=URxforanryx2X+impliesgË=e.BItiseasytovrerifyV,thataG-setXisaprincipalhomogeneousspace6i themap'UT:cX :Go3(x;gn9)7!(x;xgn9)2X :X:isbijectivre.+ This6holds97alsointhecaseXFչ=UR;.IfX6=UR;thenX*andGareisomorphicas6G-sets.8Thesestatemenrtsareeasilytranslatedintotermsoffunctors.BThemap'UT:X "GURn!1XXwhicrhisde nedforanyG-setvXalued6functor Xinducesthealgebrahomomorphism :A5# A3s t7!6टPEOst(A)3Q t(H$q% cmsy6)"2yA HV2 ontherepresenrtingobjects.w'isanisomor-6phismi Xis.6Prop`ositionc>2.4.B)LffetOfGbearepresentablegroupvaluedfunctorandX6bffeBarepresentableG-setvaluedfunctoronkg-RAlgfl.Lettherepresenting6algebrffaaAofXSzbefaithfully at.ThenGandXSzareKܞ-formsofeach6otherasG-setsforsomefaithfully atcffommutativekg-algebraK,i X6is35aprincipffalhomogeneousspaceoverG.6Prffoof.ZWVe rstremarkthefollorwing.LetXandYSbSerepresentable6functors,4letfS: X!BYUbSeanaturaltransformation,andKbSefaith-6fully, at.+lAssumethatfGjK :*=XjK =0V! X+YpjK !isanisomorphism.Then6f 9is:anisomorphism.ThisisduetothefactthatthecorrespSonding6statemenrtholdsfortherepresentingalgebras.BNorwKlettherebSeanaturalisomorphismofGjK;¹-setvXaluedfunctors6 h:bGjK u!XjK;¹.Then`since(X"0Yp)jK =(XjK lYYpjK "the`follorwing6diagramcommrutes?  (GG)jK (GG)jK`:2fd!ά-˰M'jKKi(X+G)jKȹ(X+X)jK32fdά-c'jK(Ǡ@feZǠ?ꬾ 1(Ǡ@feZǠ?ꃀ  6 :DWBहSincevGjK 8isaprincipalhomogeneousspaceorvervGjK"compSonen-6trwise",the{topmorphismisanisomorphism.Soarethetwovertical6arrorws. \ThusLthebSottomarrorwisanisomorphism.BytheabSorve6argumenrtwegetthat'UT:X+GURn!1XX+isanisomorphism.BConrversely;if'UT:XqG߃!ؓXX-[is;anisomorphism,P$theninpar-6ticulartheinducedk-algebrahomomorphism Í: zAr AL3sr tL7!6टPEOst(A) plt(H)N2CA HV2 zof therepresenrtingalgebrasisanisomor-6phism.?*(HereBwreusetheSweedlernotationincontextwithabilinear gѠo$pIF9ORMS!OFHOPFALGEBRASANDGALOISTHEOR:Y49V6हmap.)@ThisrrisevrenanisomorphismofA-algebras.Sowregetforany6A-algebraB;a?GjA(B)PUR԰n9=kg-QAlgfl(HV Z;B)PUR԰n9=A-Algfl(A HV Z;B)P԰쳹=~uA-Algfl(A A;B)PUR԰n9=kg-QAlg(A;B)PUR԰n9=XjA(B):6हItJisnorweasytoverifythatthisisanisomorphismofGjA-setvXalued6functors.1( msam10N%BहThe]translationofthenotionofprincipalhomogeneousspacesinrto6termsofHopfalgebrashasamostinrterestingvXariation. bLetAbSe6an HV2Z-comoSdulealgebra.AssumenorwthatHV2 Wis nitelygenerated6andmprojectivreasakg-moSduleandthatAisfaithfully at./Thedual6H:=MHomD֟k#hh(HV2Z;kg)\isa nitelygeneratedprojectivrecoScommutative6Hopfw$algebrawhicrhactsonAbyht0=Pt(A) z\h(t(H) ). VThenw$the6follorwingholds:č6Theorem1andDe nition2.5.WUndertheabffoveassumptionsthefol-6lowing35arffeequivalent:MRda)\$ABisaHopfGaloisextensionofkK_withHopfalgebrffaHј(or\$simply35HV-Galois).LHb)\$ Ë:URAʸ A3s t7!Pst(A)E t(H)`b2A HV2 is[;anisomorphism.Mc)\$Therffeisafaithfully atextensionK]'ofkwithK< APUR԰n9=K HV2\$as35KF HV2Z-cffomodulealgebras.LHd)\$ֹ:HR A3h s7!(t7!P-Os(ht))2Endck (A)iisan\$isomorphism)BandAis nitelygenerffated)Bfaithfulprffojective)Basa\$kg-moffdule.Me)\$kRis35the xringhA H n:=URfs2Aj8h2HB:hsUR=(h)sg;\$of3AundertheactionofH andtheringsA2H LandA#Harffe\$Morita35effquivalent.N%6Prffoof.Za):1AHopfJGaloisextensionisde nedtobSeoneoftheequivXalenrt6conditionsl$b)-e). Ub)impliesc)withK=vA.TheequivXalenceof6b)andc)istheprecedingPropSosition.~pTheequivXalencebetrweenb)6andhd)isasimplecalculationwithdualbasesforHUandHV2 anduse6offaithful atness.!e)isessenrtiallyatranslationofd)intotermsof6MoritaequivXalences.8DetailedproSofsofthiscanbefoundin[P].-PN%BहThere|arevXariousdi erenrtgeneralizationsofGaloisextensions.|Non-6commrutative algebraswithHopfalgebrasactingonthemharve bSeen6inrvestigated.++Commutativealgebraswith nitegroupsactingonthem xNo$610yBODOTP:AREIGISV6हharve[bSeenstudiedin[CHR].Thede nitionusedherehasbeeninrtro-6ducedin[CS]andisalsodescribSedin[S1].5LSpecialinstancesofGalois6extensionsareencludedinthisgeneralconcept.BLetnk&bSea eldandHҹ=|kgGthegroup(Hopf8)algebraofa nite6group.LetK1bSea eldextensionofkkwhicrhisHV-Galois.ThenG6हacts CbryautomorphismsonKܞ.FVurthermorewehavek=jfs2Kܞj8g26GUT:gn9(s)UR=sg=Kܞ2G6.7Since[K1:kg]=jGjwregetthatK!isa"classical"6Galois extensionofk;=withGaloisgroupG.1^Conrversely ifKisa"clas-6sical"GaloisextensionofkιwithGaloisgroupGthenbryDedekind's6lemmaandd)oftheabSorveTheoremKisHopfGaloiswithHopfalge-6braHB=URkgG.BJacobson'sextension[J]ofGaloistheorytopurelyinseparable eld6extensionsFcanbSeincorporatedinrtothegeneralframeworkofHopfGa-6loistheoryinthefollorwingwayV.JacobsonusesrestrictedLiealgebras6actingbryderivXationsonpurelyinseparable eldextensionsofexpSonent6one.BThe%restrictedunivrersalenvelopingalgebrasoftherestrictedLie6algebrasareHopfalgebrasandtheactionextendstoaHopfGaloisac-6tionLonthesameextension.Detailsandanextensiontoalargerclass6ofFixQT:XHfHV 0URHVjH 0Hopf35subffalgebra\g!fLjkoLKsubffalgebra<]g6is35injeffctiveandinclusion-reversing.CBहWVesary,thatthefundamenrtaltheoremofGaloistheoryholdinits6strffongform,ifthemapFixisbijectivre. IThis,however,isnotthe6case ingeneral,.aswrewillseebSelow.ڈThereisanotherdeviationfrom6the|z"classical"GaloistheoryV. XTheHopfalgebraactingonaGalois6extension|6KXԹofkSisnotuniquelydetermined. Examplesharve|6bSeen6knorwnforinseparable eldextensions. Qo$pIF9ORMS!OFHOPFALGEBRASANDGALOISTHEOR:Y/11VV3._Sep32arableFieldExtensionsBहWVe"givreanexampleofaseparable eldextensionwhichisnotGalois6intheclassicalsense,butwhicrhisHopfGaloiswithtwodi erentHopf8M6algebras.#LetqK1=URQ(lAacmr64UUp UWljz 925S)andko=Q.#Itiswrellknownthatthisisnot6a"classical"Galoisextension.8Let緍e;HB=URQ[c;s]=(c 2j+s 21;cs)ʰ6with~thecoalgebrastructureasgivreninpartI.Abbreviate!:=l4pljz 92.6ThentheopSerationofHonKFisgivrenby+ᙚy@36ffş1j9! )t!n922(!n923ffp〄ff ʍc〟36ff! 1A0[>]!n922̹0fs〟36ff! 0:"!gHX0!n923.=:/BहIfK1=URkg(l4UUp UWljz 925S)wrereaclassicalGaloisextensionforexampleoverthe6base3 eldQ(i)thentheGaloisgroupiscyclicwithgeneratore.Here6theHgeneratorehasbSeenreplacedbrythetwoopSeratorscandswhich6opSeratekg-linearlyandaccordingtotherules緍Cjc(xyn9)UR=c(x)c(y)s(x)s(y)and2Cls(xy)UR=s(x)c(y)+c(x)s(y):6हSimilaritiesp]withthetrigonometricequalitiesareinrtended.Ifoneex-8M6tends2thebase eldfromQtoQ(i)then(Q(i)[k Q(URl4UUp UWljz 925S))UR:Q(i)2bSecomes6aclassicalGaloisextensionandtheHopfalgebraHisextendedtothe6groupringQC4.Byfurtherextendingthebase eldtoQ(i;UPl4USp UUljz 925Q)the6ringextensionQ(i;UPl4USp UUljz 925Q) Q(URl4UUp UWljz 925S)bSecomesisomorphictothedualofthe6extended@groupalgebra(Q(i;UPl4USp UUljz 925Q)C4)2.8This@isomorphismiscompati-6blewiththecomoSdulealgebrastructure.9Sowreseethattheoriginal6HV2Z-comoSdulepalgebraKMisaformofthetrivialHV2-comoSdulealgebra6HV2Z.BOnezcanshorwthatthereisasecondHopfalgebraoverQandaction8M6onwKR=vZQ(URl4UUp UWljz 925S)sucrhthatthesetupisaHopfGaloisextension.6MThe6HopfalgebraisRu=HB=URQ[c;s]=(s 2j2c 2+2;cs)t6withtheaction)30ᙚ~36ffĤ1ބ!w!n922"$!n923ffuğ〄ff ʍc〟36ff! 19 0Rb!n922t`0fs〟36ff! 06z!n923Z0x=2!n9:BहThe%Hmapscandsarekg-linearandsatisfythemrultiplicative%Hrelations>9c(xyn9)UR=c(x)c(y)ō۹1۟[z ΍2 s(x)s(y)and2Cls(xy)=c(x)s(y)+s(x)c(y):xI6हTVoWseethatthisgivresaHopfGaloisextensiononehastoextendthe6base eldtoQ(pljz5S m92)andthenproScedeasaborve. o$612yBODOTP:AREIGISVBहThisEfisanexampleofakg-algebrawhicrhisaHopfGaloisextension6withJtrwodi erentHopfalgebras.\WVewillseefurtherdownthatthis6willhappSenveryoften.bEvrenthe"classical"Galoisextensionsoften6harvemorethanoneHopfalgebraforwhicrhtheyareHopfGalois.|On6theotherhandthereareseparable eldextensionswhicrharenotHopf6Galoisu}atall.Theseparable eldextensionswhicrhareHopfGaloiscan6bSeclassi edbrythefollowingtheorem.BTVoГformrulatethetheoremwe xthefollowingnotation.0/LetK1bSea6 niteseparable eldextensionofkg.8Assume43|xs~pK=URnormalclosureof_KForverkg;2rK G=URAut=(x$~K ۶=kg);o|G20=URAut=(x$~K ۶=Kܞ);sS=URG=G20xٹ(leftcosets),rB=URPrerm(S׹)(groupofpSermrutationsofS):54h6TheoremTc3.1.[GP]HUnderQtheassumptionsmadeabffovethefollowing6arffe35equivalent?MRda)\$Therffe35isaHopfkg-algebraH suchthatK5=kRisHV-Galois.LHb)\$TherffeisaregularsubgroupN6URBsuchthatthesubgroupGUR\$B;normalizes35N@.BहTheexamplesgivrenabSoveareofaratherspSecialtypSewhichwecall6"almosttclassical"HopfGaloisextensions.Theyarecrharacterizedby6thefollorwing6Theorem3.2.[GP]35Thefollowingcffonditionsareequivalent:MRda)\$TherffeexistsaGaloisextensionE=kvsuchthatE [Kcisa elde\$cffontainingxX.~35K.TLHb)\$Therffe35existsaGaloisextensionE=kRsuchthatE^ K1=xzK~URK1.Mc)\$G20nhas35anormalcffomplementNtinG.LHd)\$Therffe=existsaregularsubgroupN:BCnormalizedbyGand\$conrtained35inG.BहThelastconditionofthistheoremshorwsthatweareindeedtalking6abSoutiHopfGaloisextensions.<$Theseextensionsareparticularlywrell6bSeharvedbecausetheysatisfythefundamenrtaltheoremofGaloistheory6initsstrongform.6Theorem&3.3.[GP]-IfK5=kisalmostclassicffallyGalois,thenthere6is> aHopfalgebrffaH+`suchthatK5=k'isHV-GaloisandthemapFixis6bijeffctive.BहTheamrbiguityoftheHopfalgebraactingonaHopfGaloisextension6isexpSosedinthefollorwing o$pIF9ORMS!OFHOPFALGEBRASANDGALOISTHEOR:Y/13V6Theoremp3.4.[GP]KGA2nyKclassicffalGaloisextensionK5=kcanbeen-6doweffd\DwithanHV-Galoisstructuresuchthatthefollowingvariantof6the:fundamentalthefforem:holds:uThereisacanonicalbijectionbetween6Hopf35subffalgebrasofH andnormalintermediate eldskoURE iKܞ.xBहOneNhofthesimplestexamplesofaclassicalGaloisextensionwithv6thisnewHV-Galoisstructureisthefollorwing.|Let bSea32rdprimitive8M6roSotOofunitryandlet!:=.l31p 3ljz 92/.^ThenKGz=jQ(!n9;)isaclassicalGalois6extension_ofQwithGaloisgroupS3.6ItisalsoHopfGaloiswithHopf6algebra@HB=URQ[c;s;t]=(c(c1)(c+1);2c 2j+st+ts2;cs;sc;ct;tc;s 2;t 2):6हTheactionofHonKFisdescribSedbrythetable9ә36ff&l1^!0Fffarӟ〄ff ʍc〟36ff! 19 0Rb22fs〟36ff! 06z!n922U3H0t〟36ff! 09 0Rf0.BTheactionofthethreegeneratingelemenrtsc;s;t꨹onKFsati es"Zʍ~c(xyn9)"=URc(x)c(yn9)+Fu1۟z@2 Qs(x)t(y)+Fu1۟z@2 Qt(x)s(y);}[s(xyn9)"=URc(x)s(yn9)+s(x)c(y)+Fu1۟z@2 Qt(x)t(y);~t(xyn9)"=URc(x)t(yn9)+t(x)c(y)+s(x)s(y):BहWVe nishthisparagraphonseparable eldextensionswhicrhareHopf6Galoisbrygivingafamilyofexamplesofseparable eldextensionswhich6areVnotHopfGalois:Gno eldextensionK3yorverVQofdegree5withT6automorphismgroupS5ofx~K^=kQŹcanbSeHopfGalois[GP].*s獍4.BHopfAlgebraFformsrevisitedBहManrycofthefollowingresultshavebSeenobtainedincooperationand6discussionswithstudenrtsandcolleguesofmine.hInparticularIgrate-6fullyd acrknowledgethecoSoperationofC.Greither,dR.Haggenmvuller,6andC.WVenninger.BThetecrhniquestoproveTheorem1.2canbSeusedtocalculatemore6formsofgrouprings.TheadvXanrtageintheproSofofTheorem1.2was6that allquadraticextensionsofacommrutative ringcanbSeexplicitly6describSedMiftheringsatis esonlyminorconditions[Sm].If2isnot6azerodivisorink&andifPic(2) \|(kg)#=0thenallquadraticextensions`6ofk&arefreeandcanbSedescribedasKM=kg[x]=(x227;3axb)where6a22:ù+z4bUR=u؉isaunitinkg.݁Thenon-trivialautomorphismisfG(x)UR=ax.6This(informationwrastranslatedintotermsofHopfalgebraformsusing6thefollorwing&o$614yBODOTP:AREIGISV6TheoremZX4.1.ѹ[HP]NLffetGbea nitelygeneratedgroupwith nite6automorphismVgrffoupF=URGdBrp W|-Aut"rv(G)Thenthereisabijectionbetween6हGal(kg;Fƹ),thesetofismorphismclassesofF-Galoisextensionsofkg,6andIHopf(kgG),thesetofHopfalgebrffaformsofkG. Thisbijeffction6assoffciates35witheachF-GaloisextensionKofkRtheHopfalgebra;WK{HB=UR n USXcggË2URKܞG33͎ UR8fQ2F: :XfG(cg)f(gn9)UR=Xcggn9 o V:<6Furthermorffe35H isaKܞ-formofkgGbytheisomorphism!Í: H KP1԰J׹=ܙKܞG;!n9(h a)UR=ah:BहOntheotherhanditisnottrivialtodescribSeFƹ-Galoisextensionsofa6 eldXkg.>TheyarenotjusttheclassicalGalois eldextensionsofk.>The6simpleJexampleofthetrivialFƹ-Galoisextensionkg2FP ˾԰ 䥹=vgkɢb:::bkisnot6a0 eld.yActuallyFƹ-GaloisextensionsarejustHopfGaloisextensions6withF\HopfalgebrakgF"[CHR,Thm1.3].KArbitrarycommrutativeF\rings6Kare=admittedasGaloisextensions.TheactionofthegroupF͹onthe6extensiond~KAbrydi erentelementsf;fG20zhastobSe"stronglydistict",Si.e.6for3evreryidempSotenteUR2Kѹthere3isanxUR2Ksucrh3thatfG(x)eUR6=f208(x)e.6Thisisthekreytothefollowing艍6Theoremo4.2.jvLffetFbea nitegroupandkYa eld.SK5=kisanF-6Galois35extensionifandonlyif$ZFKP1԰J׹=ōnotimes ;͍ܙzp}|p{ 33ܙL:::L6wherffeL=kisaU@-Galois eldextensionwithU.JF3oasubgroupof6index35n.^6Prffoof.ZLetiK5=kbSeanFƹ-Galoisextension. KFisacommrutativeisepara-6blekg-algebrabry[CHRThm1.3]henceisaproSductKP1԰J׹=ܙL1ˑ ::: Ln6हofQseparable eldextensionsLid=kg.nTheautomorphismsinFmapthe6primitivreaidempSotentstoprimitiveidempSotentsandFvopSeratestransi-6tivrely+onthesetprimitiveidempSotents,<sincethesumofidempSotents6in!anorbitisinthe xed eld.4FVoranrytwoidempSotentseiBandejD+the6automorphism0fxƹofFҍmappingeitoejѹalsomapsLitoLjf . >HenceLi6हis"isomorphictoasub eldofLjf .vNBysymmetryallthe eldsLicare6mrutually isomorphic.ThestabilizerU̓F^ofe1ʜactsasGaloisgroup6onL1=kQŹsinceitactsstronglydistinctlyandjU@jUR=[L:kg].BConrverselywletU=GbSeasubgroupandL:kJbSeU@-Galois.#MLet6g1;:::ʜ;gn JpbSe asetofrepresenrtatives forG=U΀=fg1U;:::ʜ;gnPU@g._ILet6Kz=;L:::^LrwithidempSotense1;:::ʜ;enP.4De netheactionÍ: 50G68/!ISn qAbryn9(g)(i)ϭ=juĹifgn9gidU=gjf U չtheregularrepresenrtationofGzQ6हonG=U@.WVede negn9(lCeidڹ)o:=g1؍I{(g)(i)ggidڹ(lC)eI{(g)(i).ObservrethatggidUS=<o$pIF9ORMS!OFHOPFALGEBRASANDGALOISTHEOR:Y/15VL6gI{(g)(i)U%impliesugI{;i1۹:=gn91؍I{(g)(i)gn9gii2U@.YyThenthe xringofK#under6the4{actionofGiskg,FforletP&lidei72Kܞ2G6.YThenforallgA32GwrehaveQ6टPEOugI{;i L(lidڹ)eI{(g)(i)=nPeliei.(FVorg:=ngiugn91 iZwregetgn9giU=ngiU@,Qhence+6n9(g)(i)UR=iäandugI{;i k=URgn91 i ʵgidugn91 igi,=u,qsoäthatu(lidڹ)=li(~forallu2U@,zQ6hence!lin2 kg.:MFVorgx:=gjf gn91 iֹwregetgn9gidUJù=gjf U,hencen9(g)(i)=j6हandugI{;i1ܹ=gn91 j ʵgjf gn91 igij=id4,8>sothatlidej =ljf ej,8>hencelij=lj [forall6i;jӹ.This,shorwsPנlidei+=-Pei=-2kg.Obrviously,allelementsofk6हremain( xedundertheactionofGsothatk4=RKܞ2G6.bFVurthermoreK6हisNseparablebryde nition.eTVoshowthatGopSeratesstronglydistictly6it=sucesto ndforevrerygP2G;g6=idand=eiG\2K9andx2K9sucrh6thatڒgn9(x)ei,6=URxeidڹ.3Assume rstthat(g)(i)UR6=i.3ChoSoseڒx=eidڹ.ThenL6gn9(eidڹ)ei=eI{(g)(i)ei=06=ei=eidei.Ifn9(g)(i)=ithengn91 i ʵgn9gi2U`ڹand6u6=idsincegn9not=idM.LChoSoseanlƦ2Lwithu(lC)6=l߁andx=lCeidڹ.6Thengn9(x)eiJ{=g(lCeidڹ)ei=g1 i ʵggidڹ(lC)eiu(l)eiJ{6=lei=leidei=xei.mThis6concludestheproSof.v (BहObservre(bythewaythatkgC2,hasnonon-trivialforms,sinceC2has6trivialMUautomorphismgroup,fsothecorrespSondingGaloisextensionof6a4formmrustbSekQitself.Alreadythenextsimplestcasesafterstudying6the cformsofkgZ,kC3,kC4,and ckC6gcauseunsatisfactorycalculations.6WVediscussthecaseofQC5.BThe1PautomorphismgroupofC5 TisC4whicrhhasexactlyonenon-6trivialsubgroupC2.. TheC4-GaloisextensionsKnUofQcanbSeofthe6follorwingformsj3MRd1)\$KFisaC4-Galois eldextensionofQ,MRd2)\$KP1԰J׹=ܙLL꨹whereLisaquadratic eldextensionofQ,MRd3)\$KP1԰J׹=ܙQQQQ.BTheproblemisnorwtodescribSeasexplicitlyaspossibleallC2-resp.6C4-Galoisg eldextensionsKCofQ, todescribSetheactionofC4 ' onK6हand%%thencalculatetheformsaccordingtoTheorem4.1.YAssoSciated6withaC4-Galois eldextensionKFisthefollorwingformofQC5: \fHPB԰[=QQ[a]=(a 5j5pa 3+(5(p 23qn9)10'sp 'sz3 ؍q(p2j4q)?)a)6HereKsXisthesplitting eldofx24+px22+qandlplz3 Bqn9(p2j4q)K`W2-k6हnecessarilyholdsifKisaC4-Galois eldextension. Thediagonal6mapscanbSedescribedbry"6(a)UR=ō!l 1[z? ΍uvn9(u2jv2.=)2Ebù((u 5+vn9 5.=)(a a)(u 3+vn9 3)(a b+b a)+(u+vn9)(b b));Ύo$616yBODOTP:AREIGISV6हwhere!t\auUR=簏sUT簏zP~Oq33p+qpqz$: gp2j4q33ozN ΍$T2orand>vË=簏sUT簏zQ MOq33pqpqz$: gp2j4q33ozN ΍$l2`u: BहThe caseofKPv԰=f L%L withquadraticextensionL=Qissomewhat6easier.8WVegetCByHPB԰[=QQ[a]=(a 5j+5p(a 3+pa))with6.(a)UR=u 1 \|(1;1)(a a):6हwhereLUR=Q(u)isthesplitting eldofx22jp.BFinallythecaseofKP_԰x~=7Q#QQQQleadstothetrivialform6QC5.BThe*othersimpleexampleisthatofformsofQC2B>C2.Theauto-6morphism%groupofC2C2役isthesymmetricgroupS3.Norwwehave6tostudythedi erenrtcasesofS3-Galoisextensions!MRd1)\$KFisanS3-Galois eldextensionofQ,MRd2)\$KP1԰J׹=ܙLL꨹whereLisaC3-Galois eldextensionofQ,MRd3)\$KP1԰J׹=ܙLLL꨹whereLisaquadratic eldextensionofQ,MRd4)\$KP1԰J׹=ܙQQQQQQ.BInthe rstandsecondcasewregetCHB=URQ[a]=(a(a 3j+ua+vn9))6wherexQKTisthesplitting eldofx23 +ux+v抹irreducible.IfxQD=UR4u236ह27vn922 isthediscriminanrtthen;(a)UR=FuU1z5D \[{2u(3vn9a au(a c+c a)){+Fu33v33zm ꍐ2s(4u22b b+9c c+6u(b c+c b))9vn922.=(a b+b a)]ፑ6whereԍɨbUR=ō4[zm ΍v %a 3j+ō4u۟[z ΍9va+36andHcUR=ō334u33[z ΍9v]a 3j+2a 2ō۹4u22۟[zO ΍va a4u:BहInthethirdcaseofKP1԰J׹=ܙLLL꨹wregetCHB=URQ[a]=((a 2j1)(a 2u))6whereListhesplitting eldofx22juandthediagonalisG(a)UR=ō1[z ʡ ΍u2ju(WG-X(u 2ju1)a aa 3 a 3+a 3 a+a a 3G.:ՍBहTheucaseofKP1԰J׹=ܙQpQQQQQuleadstothetrivialform6Q(C2jC2).o$pIF9ORMS!OFHOPFALGEBRASANDGALOISTHEOR:Y/17V6Problems4.3."TheVgeneratorsofHopfalgebraformsandtheirdiag-6onalsU^areratherarbitraryV.yItoftenturnsoutthateitherthediagonal6ortheidealtobSefactoredoutcanbecrhosentoberelativrelysimple,6butn4notbSoth.ÄIsthereacanonicalcrhoiceofthegeneratorsofaHopf6algebraform?JY1԰W=Ok(n):::6kg(nP).26Problemsn4.5.bIt~wrouldbSeinterestingtoknowwhichgroupalgebras6orver Qharve absolutelysemisimpleforms.[TheHopfalgebraQSn Yis6itselfEabsolutelysemisimple.jTherearealsoexamplesofgroupsGwhose6groupalgebrasharvenoabsolutelysemisimpleforms.#vv5.Sep32arableHopfGaloisExtensions6Problems5.1.t=InKpartISIIwreKhaveseenexamplesofseparable eld6extensionsHe,shorwsthattheHopfalgebraHrisneveruniqueifGis6cyclicofoSddprimeporwerorder.+Hisnevreruniquefornon-abelianG.6ThisNneedsadi erenrtproSof,ghowever,thanNgivenin[C2].e}Childsalso6shorws]thatHisuniqueifGiscyclicofprimeorder,9aresultwhichwe6willextendbSelorw.2BAssume_7thatwrehavethesamesetupK5=kg;x$~K ۴=k;G;G209;S ;B=as_7inISII.6In[C2]thefollorwingresultisshown.o$pIF9ORMS!OFHOPFALGEBRASANDGALOISTHEOR:Y/19V6Prop`osition5.2. GnmnormalizestherffegularnmsubgroupNQofB si Gis6a35subffgroupoftheholomorphHol%(N@)UR=Nt3)Autm(N).UBहWVeextendTheorem2of[C2]asfollorws:6TheoremYr5.3.`Lffet*K5=kHGbeaseparable eldextensionofdegree[Kt:6kg]UR=p35aprime.fiThefollowingarffeequivalent:MRd1)\$K5=kRis35HopfGalois.MRd2)\$K5=kRis35almostclassicffallyGalois.MRd3)\$G35issolvable.BIf[aany(andall)ofthesecffonditionsholdthentheHopfalgebraHHis6unique35forK5=kRHV-Galois.+]6Prffoof.ZIf!K5=k>isHV-GaloisthenthereisaregularsubgroupN[ofSp6हsucrhthatG+Hol((N@)=N+3dfйAutN(N),DtheholomorphofN.uSince6NP6԰=@CptheholomorphHol(N@)PUR԰n9=Cp3\Cp1ٹ,henceGissolvXable.BLetGbSesolvXable.SinceK= =`lkg(a)withaazeroofanirreducible6pSolynomial fofdegreep,GisasubgroupofSp],hencep=jGjand6p22 = jGj.aBelorw5weshowthatforasolvXablegroupGthereisachain6ofnormalsubgroupsofG(!)G9e/:::/G2j/G1/G0V=URG6हwithGid=Gi+1P԰=AQ(Z=piZ)2e8:i.8Considerthesequenceofsub eldsVoko=URk0Vk1:::uDkm16km Z=xzK~K1;6हwithski+1AV=kiwMGaloiswithGaloisgroup(Z=pidZ)2e8:i Uandki+1=kynormal.e6WVegeta6m2km yanda=2lkm1@,otherwisex~Kkm1|sincekm1@=k6हnormal.Then5theminimalpSolynomialf}ofaisirreducibleorver5km16हbrytthelastlemma.ESoallthekg-generatingelements1;a;:::ʜ;a2p1|Mof6Kare>linearlyindepSendenrtoverkm1@.5ThusKandkm1 5arelinearly6disjoinrt:andpm 8=tp.(Sincep22 =R[x$~K*:kg]andp=pm 8=[km:km1@]:wree6getkm15 T?KP԰ѹ=,km1K=Lkm =x"E~K.$#SobryTheorem3.2the eld6extensionK5=kQŹisalmostclassicallyGalois.BTVo:seethattheHopfalgebraH'togetherwiththeGaloisopSerationis6uniquely(determined,uobservrethatp=jGjandGURN+3`AutG(N@)(andN6हtheConlySylorwp-subgroupofHol5r(N@)implythattheSylowp-subgroup6ofGisN@,>whicrhisunique.7ThusGv̹=N+3JrAwithasubgroupAv6हAutJi(N@).ConsequenrtlyFNܹ=GpjUBLisuniquelydeterminedandsois6HbryTheorem3.1.O+]BहTVo nishtheproSofofthetheoremwreprovethefollowinglemmas.U6Lemma5.4.gLffet|mGbea nitesolvablegroup.)|ThenthereisasequenceG9e/:::/G2j/G1/G0V=URG o$620yBODOTP:AREIGISV6with35Gid=Gi+1P԰=AQ(Z=piZ)2e8:i,piprimeandGi/Gnormalsubffgroups.6Prffoof.Zbryinduction./WVeonlyindicatehowtoconstructGi+1OfromGidڹ.6Let/MZ4/PGi bSeanormalsubgroupofprimeindexpidڹ. vDe neGi+1L:=6टT@টgI{2GSgn9M@g21 ʵ.8IthasalltherequiredpropSerties.j">6Lemma5.5.LffetϞK5=k6beanormalseparable nite eldextension.E7Let6f"2#kg[x]]:bffeseparableandirreducibleofdegreepaprime.zTheneither6f{4is35irrffeducibleoverKorf{4completelysplitsintolinearfactors.6Prffoof.ZLeta1;:::ʜ;apbSethezerosoffZinthealgebraicclosureandletG6हbSe/theautomorphismgroupofKܞ(a1;:::ʜ;ap])=kg. Goperatestransitivrely6on~thezeros, sincef }isirreducible.2LetNbbSethe xgroupofKܞ.Since6K5=kis>normal,a$wregetthatNq/KGisanormalsubgroup.NdecompSoses6fa1;:::ʜ;ap]ginrtoorbitsofequalcardinalitysinceGopSeratestransitively6andN1|isnormal.JSoeitherNopSeratestransitivrelyortriviallyV.JHence6f2isirreducibleorsplitscompletelyV.&Xo$pIF9ORMS!OFHOPFALGEBRASANDGALOISTHEOR:Y/21VReferences6ल[C1]M2p0J cmsl10L.4Childs:T*amingWildExtensionswithHopfAlgebras. KnT*rans.UUAmer.Math.SoGc.304,p.111-140.1987.6[C2]ML.ˊChilds:^1OntheHopfGaloisTheoryforSeparableFieldExtensions.toKnappGearUUComm.Alg.6[CHR]W5S.Chase,*D.K.Harrison,A.RosenbGerg:GaloisTheoryandGaloisCoho-KnmologyUUofCommutativeUURings.Mem.Am.Math.SoGc.52(1965).6[CS]N5S.Chase,:M.Sweedler:HopfAlgebrasandGaloisTheory*.LNinMath.97.KnSpringerUU1966.6[GP]PC.-Greither, hB.Pareigis:PHopfGaloisTheoryforSeparableFieldExtensions.KnJ.UUAlgebra106,p.239-258,1987.6[G]KnA.Grothendieck:T*echniquededescenteI.SeminaireBourbaki,EExp.190.Kn1959/60. ԍ6[H]KnR.HaggenmGuller:kx+UbGerInvqariantenseparablerGaloiserweiterungenkommuta-KntiverUURinge.DissertationUniversitatMGunchen1979.6[J]KnN.QJacobson: AnExtensionofGaloisTheorytoNon-NormalandNon-KnSeparableUUFields.Am.J.Math.66,p.1-29.1944.6[KO]PM.-A.lKnus,qM.Ojanguren:CTheoriedelaDescenteetAlgGebresd'Azumaya.KnLNUUinMath.389,Springer1974.6[P]KnB.Pareigis: DescentTheoryappliedtoGaloisTheory*.TechnicalRepGortKnUniv.UUCalifornia,SanDiego,1986.6[Sm]ORoC.HSmall:ATheGroupofQuadraticExtensions.J.PureAppl.Alg.2,p.83-105,Kn395.UU1972.6[S1]KM.UUSweedler:qHopfAlgebras.Benjamin{NewY*ork1969.6[S2]KM.ESweedler:ŧStructureofPurelyInseparableExtensions.Ann.Math.87,Knp.UU401-411.1968.6[W]KC.PH.W*enninger: Hopf-Galois-TheorieeinerKlassereininseparablerKnKorpGererweiterungen.UUDiplomThesisUniversitatMGunchen1984.B3- cmcsc10MaUTthematischesInstitutderUniversitM*atMM*unchen,GermanyBE-mailaddr}'ess!:q4