◆ Thecoverpageusesthefonts BebasNeue and LeagueGothique,bothlicensed underthe SILOpenFontLicense.
BACKGROUNDS
Throughoutthis course,thereader is assumedtohave acquaintancewithundergraduatealgebra,namelythebasicnotionsaboutsets,groups,ringsandmodules.Details canbefoundinanydecenttextbooksuchas[11].Inordertorecalltherelevantnotions (inEnglish!)andto xthenotations,wegivearecapitulationbelow.
Sets WeworkintheframeworkofZFCsettheory.Theusualoperationsonsetsare: ∩, ∪, ×, ⊔ (=disjointunion);theCartesianproduct(resp.intersection,union, disjointunion)ofafamilyofsets {Ei : i ∈ I} isdenotedby i∈I Ei (resp. i∈I Ei, i∈I Ei, i∈I );thesetofmapsfrom X to Y aredenotedby YX.Thecardinality ofaset E isdenotedby |E| or#E.Forthemostpartinthiscourse,weneglect set-theoreticissuessuchasproperclasses,etc.
If f : X → Y isamapand E ⊂ Y,wewrite f 1 (E ) : {x ∈ X : f ( x ) ∈ E};when E {y} weusetheshorthand f 1 ( y ) f 1 ( {y} ) ,commonlycalledthe ber of f over y.
Thesymbol A : B readsas“A isde nedtobe B”.Thearrow ֒→ (resp. ։)means aninjection(resp.surjection),and x → y meansthattheelement x ismapped to y.If ∼ isanequivalencerelationonaset E,thecorrespondingquotientsetis denotedby E/ ∼
Group Agroupisaset G endowedwithabinaryoperation(“multiplication”) ( x , y ) → x · y xy,suchthat
⋆ associativityholds: x ( yz ) ( xy ) z,sothatonemaysafelywritethemas xyz;
⋆ theunitelement1exists: x · 1 1 · x x forall x —werefrainfromthe commonbutawkwardsymbol e fortheunit;
⋆ everyelement x ∈ G isinvertible:thereexists x 1 ∈ G,necessarilyunique, suchthat xx 1 x 1x 1.
Ifweremovetheexistenceofinverses,thestructureso-obtainediscalleda monoid When G iscommutative/abelian,i.e. xy yx holdstrueforall x , y ∈ G,itis
customaryto writethegroupoperationsintheadditivemanner: x + y,0, x insteadof xy,1and x 1,respectively.Inthiscasewesay G isanadditivegroup.
Thenotation H ⊳ G meansthat H isanormalsubgroupof G.Thesymmetric groupon n letters,say {1,..., n},willbedenotedby Sn.
Rings Unlessotherwisespeci ed,theringsareassumedtohavemultiplicativeunit element1.Therefore,aring R isanadditivegroup ( R, +, 0) togetherwithamultiplicationmap ( x , y ) → x · y xy thatmakes ( R, · , 1) intoamonoid.These structuresarerelatedby distributivity:
x ( y + z ) xy + xz , z ( x + y ) zx + zy.
Thestandardexampleforacommutativeringistheringofintegers ,thenoncommutativecaseisbestillustratedbytheringof n × n-matrices.Anelement x ∈ R iscalled invertible (ora unit of R)if ∃y ∈ R, xy yx 1;inthiscase y isuniqueandwedenoteitas x 1.Theunitsformagroupundermultiplication, denotedas R× .
Fields When R {0} F×,wecall R a divisionring;a eldisacommutativedivision ring.Weshallwrite , , forthe eldsofrational,realandcomplexnumbers.The nite eldwith q elements(q:aprimepower)isdenotedas q.
Let F[X]standforthe ringofpolynomials intheindeterminate X withcoe cients in F.Its eldoffractionsisdenotedby F ( X ) ,calledtheringof rationalfunctions in X.Byconstruction, F ( X ) consistsofquotients P/Q where P, Q ∈ F[X]and Q 0. Likewise,onecande netheirmultivariateavatars F[X, Y,...]and F ( X, Y,... ) Wewillalsoencounterthebroadercaseofthepolynomialring R[X, ]overa commutativering R.
Homomorphisms aremapsthatrespectalgebraicstructures,namelytheconditions suchas ϕ ( xy ) ϕ ( x ) ϕ ( y ) and ϕ (1) 1areimposed.Forahomomorphism ϕ between groups(resp.rings),wedenoteits kernel and image byker ( ϕ ) : ϕ 1 (1) (resp.ker ( ϕ ) : ϕ 1 (0) )andim ( ϕ ) .
Thereisalsoanotionof“substructures”,namelythesubgroups,subrings,etc.A subgroup N ⊂ G iscalled normal if xNx 1 ⊂ N forall x ∈ G,inwhichcasewewrite N ⊳ G.Toanormalsubgrouponeassociatesthe quotientgroup G/N
Asforrings,itturnsoutthatthetwo-sided ideals playarôlesimilartothatofnormal subgroups.Let R bearing.Anadditivesubgroup I of R iscalleda(two-sided)idealif xI ⊂ I and Ix ⊂ I forall x ∈ R.Forcommutativeringsonemaysimplyspeakofideals, withoutspecifyingthesides.Thequotientring R/I istheadditivequotientgroup R/I equippedwiththemultiplication ( x + I )( y + I ) xy + I.
Thetwo-sidedidealgeneratedbyelements x1 ,..., xn ∈ R willbewrittenas
We beginbyreviewingtherudimentsof eldtheory.Anyring R admitsexactlyone homomorphismfrom ,namely
R a −→ a · 1.
Itsimagemustbeoftheform /p forauniquelydeterminedinteger p ≥ 0.Assume that R hasnozero-divisors,i.e. xy 0 ⇐⇒ x 0 ∨ y 0,thensois /p ,andone concludesimmediatelythat p iseitheraprimenumberorzero.
De nition1.1.1. Let F bea eld.Its characteristic isthenumber p above.Denoteitby char ( F ) .
De nition1.1.2. Anintersectionofsub eldsof F isstillasub eld,thusitmakessense totalkaboutthesmallestsub eldinside F.Callitthe prime eld of F.
⋆ eitherchar ( F ) 0,inwhichcase ֒→ F andweobtainacopyof inside F by invertingthenonzerointegers;
⋆ orchar ( F ) p > 0,inwhichcaseweobtainacopyof /p : p inside F.
Summingup,theprime eldof F is or p,accordingtowhetherchar ( F ) iszeroora primenumber p. Nextcomesthenotionof compositum.
De nition1.1.3. Let F, F′ betwosub eldsofanambient eld L.Theircompositum, writtenas FF′,isthesmallestsub eldof L containingboth F and F′.Moreconcretely, theelementsof FF′ taketheform x1x′ 1 + ··· xn x′ n y1 y′ 1 + ··· ym y′ m ∈ L
with xi , yi ∈ F, x′ i , y′ i ∈ F′ suchthatthedenominatorisnonzero. Likewise,thecompositumofanarbitraryfamilyofsub eldsinside L canbede ned.
Notethataringhomomorphismbetween elds ϕ : F → E musthavekernelequal to {0}.Thus,insteadoftalkingabouthomomorphisms,onemayconcentrateon embeddings ofa eld F intoanother eld.If E ⊃ F,wesaythat E isa( eld)extensionof F;it iscustomarytowritesuchanextensionas E/F —donotconfusewithquotients!Field extensionswillbethemainconcernofthislecture.
Let E/F beanextension.Notethat E formsan F-vectorspace:theadditionin E and thescalarmultiplicationof F on E comefromtheirringstructures.
De nition1.1.4. The degree of E/F isde nedasdimF E,alsowrittenas[E : F].Extensionsof nitedegreearecalled niteextensions
Here[E : F]isregardedasacardinalnumber.
Lemma1.1.5 (Towerproperty). If F ⊂ E ⊂ L are elds,then [L : F] [L : E][E : F]
ascardinalnumbers.Inparticular, L/F is niteifandonlyif L/E and E/F areboth nite.
Proof. Chooseabasis B (resp. C)ofthe F-vectorspace E (resp.ofthe E-vectorspace L). Everyelement v ∈ L hasabeuniquelyexpression v c∈C γ
c ( nitesum),γc ∈ E.
Expandingeach γc asan F-linearcombination
,wearriveataunique expression
Thisprovidesabasisfor L whichisinbijectionwith B × C,provingourassertions.
1.2Algebraicity
Theinnocent-lookingnotionof nitenessisdirectlyrelatedto algebraicity,asreviewed below.Consideranextension E/F.Foranyelement u ∈ E,wewrite F ( u ) asthesub eld generatedby u,thatis:
( u )
Itselementscanbeexpressedas P ( u ) /Q ( u ) ,where P, Q ∈ F[X]arepolynomialsand Q ( u ) 0.Ontheotherhand,wedenote
whichisasubringof F ( u ) .Furthermore,wemayallowmorethanonegenerators u ,...,,andobtain F ( u ,... ) and F[u ,...]intheevidentmanner.
Theelement u issaidtobe algebraic over F,ifthereexistsanonzeropolynomial P ∈ F[X]suchthat P ( u ) 0.Non-algebraicelementsarecalled transcendental.When E ⊃ F,werecoverthefamiliarnotionofalgebraicnumbers.
Lemma1.2.1. If u ∈ E isalgebraicover F,thereexistsanirreduciblepolynomial P ∈ F[X], uniqueuptomultiplicationby F×,suchthat [Q ∈ F[X], Q ( u ) 0] ⇐⇒ P|Q
Wemaynormalize P sothat P is“monic”: P ( X ) Xn + an 1Xn 1 +··· a0.Callitthe minimal polynomial of u
Proof. Let P beapolynomialsatisfying P ( u ) 0withlowestpossibledegree.Itmust beirreducible.If Q ( u ) 0,Euclideandivisionfurnishes R ∈ F[X]withdeg R < deg P and P|Q R.Theminimalityofdeg P thusimplies R isthezeropolynomial.
Conversely, niteextensionsmaybeconstructedbytakinganirreducible P ∈ F[X] andformthequotientring F[X]/ (P ) ,whichisa eldandcontains F F · (1+ (P )) . Indeed,theirreducibilityof P impliesthat F[X]/ (P ) isa eld,byastandardresultin algebra.
Proposition1.2.2. Let u ∈ E beasabove.Then u isalgebraicover F ifandonlyif F ( u ) /F is nite;inthiscase F ( u ) F[u] andthereisaringisomorphism (1.1)
F[X]/ (P ) ∼ −→ F ( u ) Q −→ Q ( u ) , where P istheminimalpolynomialof u.Inparticular, [F ( u ) : F] deg P
Proof. Assume u algebraicandlet P Xn + an 1Xn 1 + ··· + a0 bethemonicminimal polynomialof u.Weclaimthateveryelementin F[u] {0} isinvertible,andtherefore F ( u ) F[u].Indeed,if Q ( u ) 0,then P ∤ Q andirreducibilityof P togetherwiththe Euclideandivisionentailthat
1 PU + QV
forsome U, V ∈ F[X].Evaluationat u furnishes Q ( u ) V ( u ) 1,whenceourclaim.It followsthatthehomomorphism(1.1)issurjective.ThepreviousLemmaimpliesthe injectivityof(1.1).Therefore F[X]/ (P ) ∼ → F ( u ) .
Conversely,if F ( u ) /F is nite,thentheremustbean F-linearrelationbetween1, u , u2 ,... whicha ordstherequiredalgebraicequationfor u
Ontheotherhand,thestructureof F ( u ) inthetranscendentalcaseissimpler—it isjustthe eldofrationalfunctions.
Proposition1.2.3. Anelement u ∈ E istranscendentalover F ifandonlyif
( X ) −→ F ( u )
de nesaringhomomorphism,inwhichcaseitisactuallyanisomorphism.
Proof. Since F ( u ) consistsofthe“rationalfunctions”in u,themapwillbeasurjective ringhomomorphismprovidedthatitiswell-de ned,whichisinturnequivalentto that R 0 ⇐⇒ R ( u ) 0forany R ∈ F[X].Thelastconditionisclearlyequivalentto thetranscendenceof u over F;italsoimpliesthat F ( X ) → F ( u ) isinjective,thusisan isomorphism.
Proposition1.2.4. Let E/F beanextension.If α,β ∈ E arealgebraicover F,then
Proof. Considertheextensions F ⊂ F ( α ) ⊂ F ( α,β ) .Theelementslistedaboveallbelongto F ( α,β ) .Notethat β isalgebraicover F ( α ) (ofcourse,enlargingthebase eld preservesalgebraicity).ByProposition1.2.2,both[F ( α ) : F]and[F ( α,β ) : F ( α ) ]are nite.TheassertionfollowsfromLemma1.1.5
Notethatthisisjustanabstractresult: itisnotsoeasytodeterminetheminimal polynomialof α + β,etc.inpractice.Anotherconsequenceisthatthealgebraicelements in E formsasubextension Ealg/F.
Exercise1.2.5. Determinetheminimalpolynomialofthealgebraicnumber √2+ √3 over
Anextension E/F iscalledalgebraicifeveryelement u ∈ E isalgebraicover F
Proposition1.2.2impliesthat E/F isalgebraicifandonlyifitisaunionof niteextensionsof F
Exercise1.2.6. If L/E and E/F arebothalgebraic,then L/F isalgebraicaswell.
Attheotherextreme,givena eld F andapossiblyin niteset Γ,wemayform the eld F (Γ) ofrationalfunctionswithindeterminatesin Γ;when Γ {X1 ,..., Xn } werecoverthefamiliar F ( X1 ,..., Xn ) .Unlikethealgebraicsetting, Γ is algebraically independent:therearenonon-trivialpolynomialrelationsamongelementsin Γ.Itcan beshownthatevery eldextension E/F hasadecomposition
algebraic
forsomealgebraicallyindependentsubset Γ ⊂ E;moreoverthecardinalityof Γ is uniquelydeterminedby E/F,calledthe transcendencedegree of E/F.Thisshouldbe comparedwiththethenotioofbasesanddimensionsinlinearalgebra.
1.3Thealgebraicclosure
De nition1.3.1. A eld F iscalled algebraicallyclosed ifeverypolynomial P ∈ F[X]has arootin F.Thisisequivalenttothatevery P ∈ F[X]splitsintolinearfactors(i.e.of degreeone).
Equivalently,beingalgebraicallyclosedmeansthatevery P ∈ F[X]factorsintolinearfactors: P ( X ) deg P i 1 ( X ai ) where ai ∈ F.Thebestknownexampleis Henceforthwe xour“ground eld” F andstudyvariousextensionsthereof.Let E/F, E′/F betwoextensions,an F-embeddingisanembedding E → E′ of eldswhich inducesidon F.Likewisewehavethenotionof F-isomorphisms,etc.
Lemma1.3.2. Consideranextension F ( u ) /F where u isalgebraicwithminimalpolynomial P.If E/F isanextensionand v ∈ E satis es P ( v ) 0,thenthereexistsaunique F-embedding ι : F ( u ) → E suchthat ι ( u ) v.
Theorem1.3.4 (E.Steinitz). Forevery eld F,thereexistsanalgebraicclosure F of F.Moreover, F isuniqueupto F-isomorphisms.
Proof. Establishtheuniqueness rst.Let F, F′ betwoalgebraicclosures.Introducethe relation ≤ onthenonemptyset P of F-embeddings E → F′,where E/F isasubextension of F/F,bystipulatingthat
Itiseasytoseethat ( P , ≤ ) isapartiallyorderedset.WewanttoapplyZorn’sLemmato getamaximal ι : E → F′;indeed,everychainin ( P , ≤ ) hasanupperbound—simply takeunion!ByLemma1.3.2,maximalityimplies E F.
Itremainstoshowthat ι ( F ) F′.Toseethis,notethatthealgebraically-closeness of F transportsto ι ( F ) .Thisimplies ι ( F ) F′,sinceforevery u ∈ F′,therootsofthe minimalpolynomialof u over F alreadyliein ι ( F )
Asfortheexistenceof F,oneseekssomesortof“maximalalgebraicextension”of F andtheconstructionisagainbasedonZorn’sLemma.However,manipulatingthe collection(hum?)ofallalgebraicextensionsof F issomehowhazardous.Soweappeal tothefollowingdevice:thereexistsaset Ω suchthatforeveryalgebraicextension E/F, theset E isinbijectionwithasubsetof Ω.Thebasicideaissketchedasfollows.write E n≥1 En , En : {u ∈ E :[F ( u
Forevery n,themapthatassociates u ∈ En withitsminimalpolynomialover F is atmost n-to-1,soeverythingboilsdowntoboundthecardinaltiyof F[X] n≥1{P : deg P n}.
Nowweconsiderthenonemptypartiallyorderedsetformedbyalgebraicextensions E/F,where E ⊂ Ω set-theoretically,and ≤ isde nedby eldextension.Again, Zorn’sLemmaimpliestheexistenceofsomemaximal E/F.If E isnotalgebraically closed,wemayconstructanextension E′/E with ∞ > [E′ : E] > 1byLemma1.2.2.The set E′ beingalgebraicover E,thusover F byExercise1.2.6,itcanbere-embeddedinto Ω;thiswouldviolatethemaximalityof E.
Lemma1.3.5. Let K/F beanalgebraicextension,thenevery F-embedding ι : K → K isan F-automorphism.
Proof. Let v ∈ K anddenoteitsminimalpolynomialover F by P.Enumeratetheroots of P inside K as v v1 ,..., vn andset K0 : F ( v1 ,..., vm ) ,whichis niteover F.It followsthat ι inducesan F-embedding K0 → K0,whichmustbean F-automorphism fordimensionalreasons.As v isarbitrary,thesurjectivityfollowsatonce.
1.4Splitting eldsandnormality
De nition1.4.1. Analgebraicextension E/F iscalled normal ifeveryirreduciblepolynomialin F[X]splitsintolinearfactorswheneverithasarootin E
Exercise1.4.2. Let p beaprimenumberand a ∈ ≥1 whichisnota p-thpower.Show that ( a 1 p ) isnotanormalextensionof .
De nition1.4.3. Let P ∈ F[X].Anextension E/F iscalleda splitting eld for P ifthere exists n deg P roots u1 ,..., un ∈ E of P,andthat E F ( u1 ,..., un )
Moregenerally,let {Pi : i ∈ I} beafamilyofpolynomialsin F[X].Anextension E/F iscalledasplitting eldthereofifeach Pi splitsintolinearfactorsover E and E is generatedbytherootsofallthe Pi (i ∈ I)over F.
Forthestudyofsplitting eldsandnormality,itisoftenconvenienttochoosean algebraicclosure F/F.Notethatthesplitting eldinside F ofafamilyofpolynomials in F[X]istrulycanonical:simplyaddto F therootsofthesepolynomialsin F.Itis actuallythecomposituminside F ofthesplitting eldsofeach Pi.
Proof. Toprovetheexistence,we x F/F andtakethesubextensionof F/F generated bytherootsofevery Pi,asmentionedabove.
Toshowtheuniqueness,let E/F and E′/F betwosplitting eldsfor (Pi )i∈I .Take algebraicclosures E/F and E′/F;notethattheyarealsoalgebraicclosuresof F.Hence thereexistsan F-isomorphism σ : E ∼ → E′ byTheorem1.3.4.Theimage σ (E ) isstill asplitting eldof (Pi )i∈I sittinginside F′ —suchargumentisknownas transportof structure.Thereforewehave σ (E ) E′ bythepreviousdiscussionaboutsplitting elds insideanalgebraicclosure,and σ : E → E′ istherequired F-isomorphism.
Proposition1.4.5. Let E/F beanalgebraicextension,andchooseanalgebraicclosure F of E. Thefollowingareequivalent.
(i) E/F isnormal.
(ii)Every F-embedding ι of E into F satis es ι (E ) E,thusinducesan F-automorphismof E
(iii) E isthesplitting eldinside F ofafamilyofpolynomialsin F[X].
Proof. (i) ⇐⇒ (ii):Assume(i).Givenan F-embedding ι : E → F,forany u ∈ E withminimalpolynomial P ∈ F[X],weseethat ι ( u ) isstillarootof P since P has coe cientsin F.Byassumption, P splitsintolinearfactorsover E,hence ι ( u ) ∈ E andweconcludebyLemma1.3.5since u isarbitrary.Conversely,assume(ii)andlet P ∈ F[X]beirreduciblewitharoot u ∈ E.Foranyroot v ∈ F of P,Lemma1.3.2 furnishesan F-embedding E → F mapping u to v,therefore v ∈ E aswell.Itfollows that P splitsintolinearfactorsover E.Thecaseofreducible P followsatonce.
(ii) ⇒ (iii):Wecontendthat E isthesplitting eld(in F)ofthefamily {Pu : u ∈ E},where Pu ∈ F[X]istheminimalpolynomialof u.Theinclusion E ⊂ K isclear.
Conversely,the splitting eldof Pu liesin E forevery u ∈ E sincewehaveseenthat(ii) implies(i).
(iii) ⇒ (ii):Supposethat E isthesplitting eldof {Pi :∈ I} inside F.Let ι : E → F be an F-embedding,itsu cestoshowthat ι inducesan F-automorphismofthesplitting eldofeach Pi.Thisisclearsince ι permutestherootsof Pi
LECTURE2
SEPARABILITYANDFINITEGALOIS EXTENSIONS
2.1Separability
Wealways xaground eld F.If E, E′ aretwo extensionsof F,wedenotebyHomF (E, E′ ) thesetof F-embeddings E → E′.Similarly,wede nethegroupof F-automorphisms
AutF (E ) ,etc.
Considera niteextension F ( u ) /F generatedbyasingleelement u,andlet L/F be analgebraicextensioninwhich P,theminimalpolynomialof u over F,splitsintolinear factors.Recallthatwehaveestablishedthebijection
HomF ( F ( u ) , L ) ∼ −→{v ∈ L : P ( v ) 0} ϕ −→ ϕ ( u ) .
Inparticular, |HomF ( F ( u ) , L ) |≤ deg P.Strictinequalitycanholdwhenthecharacteristic p : char ( F ) ispositive.Thisleadstothenotionof separability
Lemma2.1.2. Let L/F beanextensioninwhich P ∈ F[X] splitsintolinearfactors.Then P hasmultiplerootsin L ifandonlyif (P, P′ ) 1.When P isirreducible,thelattercondition holdsifandonlyif P′ 0
Proof. Write P ∈ L[X]as n k 1 ( X ak ) with a1 ,..., an ∈ L beingtheroots.Astraightforwardmanipulationgivesthe rstassertion.When P isirreducible, (P, P′ ) 1 ⇒
P|P′ ⇒ P′ 0sincedeg P′ < deg P.
De nition2.1.3. Apolynomial P ∈ F[X]iscalled separable ifithasnomultipleroots, i.e. (P, P′ ) 1.
Weturntothestudyofirreduciblepolynomials P ( X ) k ak Xk with P′ 0.This isequivalentto kak 0forall k ≥ 1.Whenchar ( F ) 0,theonlycandidatesarethe constantpolynomials.Assumehereafterthat p : char ( F ) isaprimenumber.
Thenthepolynomials P with P′ 0taketheform P ( X ) k≥0 p|k ak Xk .
Write P P1 ( Xp ) bytaking P1 ( X )
P′ 1 0,theprocedurecanbe iteratedsothateventually
(2.1)
forsome m ∈ ≥0.Notethat P ♭ isirreduciblesince P is.Fixanalgebraicclosure F/F Itturnsoutthat {α ∈
(2.2) where βp m is the pm-throotof β in F.Infact,wehave
over F;thisisbecause p · 1 0in F andthebinomialcoe cient x y x! ( x y ) !y! satis es p | p
,
a < p , hence ( u + v ) p up + vp holdstrueinanyextensionof F. (2.3)
Iftherootsin(2.2)aretobecountedwithmultiplicities,each βp m shouldappear pm times.Summingup,thestudyofaninseparableirreducible P breaksintotwostages: (i)thestudyof P ♭ ,whichisirreducibleseparable,and(ii)thestudyof“purelyinseparable”polynomialsoftheform Xpm b
Exercise2.1.4. Thestudyofpurely inseparablepolynomialscanbefurtherreducedto thecase b Fp.Underthisassumption,showthatthepolynomial Xpm b isirreducible. Usethistoproduceexamplesofinseparable eldextensions.
Proof. Extendingtheinclusion F ֒→ F to τ : L → F isequivalentto(i)extendingit tovarious σ : E → F,andthen(ii)extendingeach E σ −→ σ (E ) ֒→ F to τ : L → F. Thereare[E : F]s choicesforthe rststep.Asregardsthesecondstep,since[L : E]s is independentofthechoiceoftheembeddingof E into F E,thereare[L : E]s choices foreach σ.
De nition-Proposition2.1.7. Let E/F bea niteextension,then[E : F]s [E : F].Call E/F a separableextension if[E : F]s [E : F].
Proof. Choose u1 ,..., un sothat E F ( u1 ,..., un ) .Usingthetower
(2.4)
andthetowerpropertiesof[E : F]s and[E : F],wereduceimmediatelytothecase E F ( u ) .Let P ∈ F[X]betheminimalpolynomialof u,andexpressitas P ( X ) P ♭ ( Xpm ) as intheearlierdiscussions,where P ♭ isseparable.Itfollowsthatdeg P [F ( u ) : F]equals [E : F]s deg P ♭ (whichisthenumberofdistinctrootsof P)times[E : F]i : pm .
Wehavejustusedtheobservationthat F ( u ) /F isseparableifandonlyif u hasseparableminimalpolynomial.Inthiscasewesay u isaseparableelement.If u ∈ E is separableover F,then u isseparableoveranyintermediate eldbetween E and F indeed,ifapolynomialhasnomultipleroots,thenthesameholdsforitsfactors.
Lemma2.1.8. A niteextension E/F isseparableifandonlyifevery u ∈ E isseparable.
De nition2.1.9. Analgebraicextension E/F iscalledseparableifeveryelementin E isseparableover F.
Exercise2.1.10. If L/E and E/F areseparable,thensois L/F
Exercise2.1.11. Suppose E is generatedbyafamily {ui : i ∈ I} over F,showthat E/F is separableifeach ui is.Henceacompositumofseparableextensionsisstillseparable.
Wesaya eld L is separablyclosed ifanyseparableirreduciblepolynomialhasaroot in L.Asinthecaseofalgebraicextensions,thereisanotionof separableclosure Fsep/F, whichisaseparableextensionwith Fsep separablyclosed.Again,wehave:
Proof. Wemayassume Fsep ⊂ F.Thenitisthesplitting eldofthefamilyofseparable irreduciblepolynomialsover F
2.2Purelyinseparableextensions
Inthissectionweassume p : char ( F ) > 0,otherwiseeverythingwouldbeseparable.
De nition2.2.1. Callanalgebraicextension E/F purelyinseparable ifeveryelement u ∈ E satis es upm ∈ F forsome m.
Weusetheshorthand E ⊂ F1/p∞ forthelastconditionde ningpureinseparability. Itmakesperfectsenseif E isembeddedintoanalgebraicclosure F and F1/p∞ istaken tobe m {u ∈ F : upm ∈ F},whichformsasub eldby(2.3).
Notethat[E : F]s 1if E ispurelyinseparable,sincewehaveobservedthatapolynomialoftheform Xpm b hasonlyonerootin F.Theassertionsbelowareimmediate.
Exercise2.2.2. If L/E and E/F arepurelyinseparable,thensois L/F.Acompositumof purelyinseparableextensionsof F isstillpurelyinseparable.
Fora niteextension E/F,wesetthe inseparabledegree tobe
Proposition2.2.3. Suppose E/F isanalgebraicextensionwith p : char ( F ) > 0.Let Es be themaximalseparablesubextension,whichmakessensebytheprecedingexercises.Then E/Es ispurelyinseparable.When E/F is nite,wehave [E : F]s [Es : F] and [E : F]i [E : Es ].
Proof. Let u ∈ E.By(2.1),there exists m ≥ 0suchthat upm hasaseparableminimal polynomial P ♭ ∈ F[X],thus u ∈ (Es ) 1/p∞ ∩ E.Weconcludethat E/Es ispurelyinseparable.Therestfollowsreadilybytowerproperties.
Exercise2.2.4. A eldiscalled perfect ifeveryalgebraicextensionof F isseparable. Showthata eld F with p : char ( F ) > 0isperfectifandonlyif F Fp : {xp : x ∈ F}.
2.3Theprimitiveelementtheorem
Theorem2.3.1 (Steinitz). Let L/F bea niteextension.Thereexistsanelement u ∈ E with L F ( u ) ifandonlyifthereareonly nitelymanyintermediate elds E (thatis, L ⊂ E ⊂ F).
Proof. Tobeginwith,weassume F nite.Thenthereareonly nitelymanyintermediate eldsbetween L and F.Ontheotherhand,awell-knownfact(eg.[11,Theorem2.18]) saysthatthe nitegroup L× iscyclic;anygeneratorof L× willthengenerate L asan extensionof F.
Assume F in niteand L F ( u ) .Foranyintermediate eld E weset PE ∈ E[X]tobe theminimalpolynomialof u over E,thus PE |PF;recallthattheminimalpolynomials arenormalizedtohaveleadingcoe cientone.Weclaimthat E E ( c0 ,... ) where c0 ,... arethecoe cientsof PE.Indeed, PE isirreducibleover E ( c0 ,... ) ⊂ E,so
[L : E] deg PE
[L : E ( c0 ,... ) ]
whichimplies E E ( c0 ,... ) .Itfollowsthatthemap {intermediate elds}−→{monicfactorsof PF in F[X]} E −→ PE isinjective.Theright-handsideis nite.
Conversely,if F isin niteandthereareonly nitelymanyintermediate elds,we maychoose u ∈ L o the( nite)unionofpropersubextensionof L,byusingthenext exercise,Then F ( u ) L
Exercise2.3.2. Let F beanin nite eld, n ≥ 1and f ( X1 ,..., Xn ) ∈ F[X1 ,..., Xn]bea nonzeropolynomial.Showthatthereexists ( x1 ,..., xn ) ∈ Fn with f ( x1 ,..., xn ) 0.
Example2.3.3. Let k bea eldwithcharacteristic p > 0.Considerthe eldofrational functions F : k ( X, Y ) intwovariables.Take x X1/p and y Y1/p insideanalgebraic closure F,andformtheextension F ( x , y ) /F.Notethat
⋆ [F ( x , y ) : F] [F ( x , y ) : F ( x ) ][F ( x ) : F] p2,and
⋆ every γ ∈ F ( x , y ) satis es γp ∈ F. Thereforethereisnoelement u ∈ F ( x , y ) suchthat F ( x , y ) F ( u ) .
Theorem2.3.4. Let E/F beaseparable niteextension.Thereexists u ∈ E suchthat E F ( u ) . If F isin niteand E F ( u1 ,..., un ) ,then u canbetakentobean F-linearcombinationof u1 ,..., un
Proof. As intheproofofTheorem2.3.1,wemayassume F in nite.Letusbeginwiththe case E F ( u , v ) .Let P, Q ∈ F[X]betheminimalpolynomialsof u and v,respectively. Wesetouttoshowthatfor“general” t ∈ F× wehave v ∈ F ( u + tv ) ,thenitfollowsthat u ( u + tv ) tv ∈ F ( u + tv ) aswell,hence F ( u , v ) F ( u + tv )
Embed F ( u , v ) into F andconsiderthepolynomials
P ( u + tv tX ) , Q ( X ) ∈ F ( u + tv ) [X].
Formtheirgreatestcommondivisor R.Since v isacommonrootof P ( u + tv tX ) and Q ( X ) in F,wehavedeg R ≥ 1.Weproceedtoshowthatdeg R 1,whichwillimply that R ( X ) X v andthus v ∈ F ( u + tv ) asrequired.
Ifdeg R > 1,thenthefact R|Q andtheseparabilityof Q wouldimplythatsome root v′ v in F isalsoarootof R.Hence P ( u + t ( v v′ )) 0,and ( u u′ ) + t ( v v′ ) 0, u u′ :rootsof P , v v′ :rootsof Q (2.5)
in F.Since F isin nite,wecanalwayschoose t ∈ F× toruleout(2.5)foranypairsof roots u u′ and v v′.Ingeneral,thisprocedureyieldsasequence v1 ,..., vn 1 ∈ L suchthat
2 , v1 ) F ( u1 ,..., un 3 )( v2 ) ··· F ( vn 1 ) and vn 1 isan F-linearcombinationof u1 ,..., un,asrequired.
ThisresultcanalsobededucedfromTheorem2.3.1bytakingtheGaloisclosure (De nition2.4.3)of E andappealtoresultsinGaloistheory,namelytheLemma2.4.5.
2.4GaloisextensionsandGaloisgroups
Let E/F beanextension,wewriteAutF (E ) forthegroupof F-automorphisms,thebinaryoperationbeingthecompositionofautomorphisms ( σ,τ ) → σ ◦ τ.Weshallwrite Aut (E ) forthegroupofall eldautomorphismsof E;itequalsAutk (E ) where k stands fortheprime eldof E,sothisisactuallyaspecialcase. Therelationcalled“transportofstructure”(afterN.Bourbaki) Autσ ( K ) (E ) σAutK (E ) σ 1 ,σ ∈ AutF (E ) (2.6) holdstrueforanyintermediate eld E ⊃ K ⊃ F. Therearetwobasicoperations.
1.Toanysubgroup H ofAutF (E ) weattachthecorresponding xed eld EH : {α ∈ E : ∀τ ∈ H,τ ( α ) α} Obviously EΓ isasubextensionof E/F.
2.Toanysubextension K/F of E weattachthesubgroupAutK (E ) ofAutF (E )
Thereoperationssatisfy
(2.7)
AutK2 (E )
De nition2.4.1. Bya Galoisextension of F wemeananormalandseparablealgebraic extension.The Galoisgroup ofaGaloisextension E/F isGal (E/F ) : AutF (E ) .
(iii)CompositaofGaloisextensionsof F arestillGalois.
De nition2.4.3. Let E/F beaseparableextension.The Galoisclosure insidesomealgebraicclosure F isthesmallestGaloisextensioncontaining E;itisgivenbythecompositumof σ (E ) , σ ∈ HomF (E, F ) .
Fixanalgebraicclosure F/F,thenthenormalityimpliesthatGal (E/F ) equalsthe setHomF (E, F ) ,andseparabilityimpliesthatthelattersethascardinality[E : F]when E/F is nite.Hence
Lemma2.4.5. Let E/F beaGaloisextension,then EGal (E/F ) F.Furthermore,themapthat sendsanintermediate eld K tothesubgroup Gal (E/K ) of Gal (E/F ) isaninjection.
Proof. Evidently F ⊂ EGal (E/F ) .Forany u ∈ EGal (E/F ) ,denoteitsminimalpolynomialby P ∈ F[X],whichmustbeseparable.If v ∈ F (achosenalgebraicclosure)isarootof P, wehaveseenthatthereisan F-embedding F ( u ) → F mapping u to v;itextendstoan element σ ofGal (E/F ) AutF (E ) .Byassumption v σ ( u ) u.Thereforedeg P 1 and u ∈ F. Sinceforanyintermediate eld K,wehaveseenthat E/K isGaloisand K EGal (E/K ) , thesecondassertionfollowsimmediately.
Noticethatthemap K → Gal (E/K ) isnotsurjectiveingeneralforin niteGalois extensions.
Lemma2.4.6 (E.Artin). Let E bea eldand H ⊂ Aut (E ) isa nitesubgroup.Then E/EH is aGaloisextensionofdegree |H|,withGaloisgroup Gal (EH /E ) H
Proof. Let u ∈ E andconsiderthe nite H-orbit O : {τ ( u ) : τ ∈ H} (withoutmultiplicities)in E.Let Pu ( X ) : α∈O ( X α ) ∈ E[X].NoticethatAut (E ) actsonthering E[X]byactingonthecoe cientsofpolynomials.Thus Pu is H- xedso Pu ∈ EH [X]; moreover Pu isseparableofdegree |O|≤|H|.Itisclearthat H ⊂ Gal (E/EH ) (2.9)
Next,weclaimthat[E : EH ] ≤|H|.Indeed,pickany u ∈ E withlargestpossible [EH ( u ) : EH ](boundedby |H|).Wemusthave E EH ( u ) ,otherwisethereexists v ∈ E withatower EH ( u , v ) EH ( u ) ⊃ EH .ByTheorem2.3.4wehave EH ( u , v ) EH ( w ) for some w ∈ E,whichcontradictsthemaximalityof[EH ( u ) : EH ].Allinall, E/EH is nite and
Theorem2.4.8 (Galoiscorrespondencefor niteextensions). Let E/F bea niteGalois extension.
(i)Therearemutuallyinversebijections
intermediate elds 1:1 ←→ subgroupsof Gal (E/F )
[E ⊃ K ⊃ F] −→ Gal (E/K )
EH ←− [H ⊂ Gal (E/F ) ], whichareorder-reversinginthesenseof (2.7)
Another random document with no related content on Scribd:
One instant in close speech
With them he doth confer: God-sped, he hasteneth on, That anxious traveller...
I was that man—in a dream: And each world's night in vain I patient wait on sleep to unveil Those vivid hills again.
Would that they three could know How yet burns on in me Love—from one lost in Paradise— For their grave courtesy.
ALEXANDER
It was the Great Alexander, Capped with a golden helm, Sate in the ages, in his floating ship, In a dead calm.
Voices of sea-maids singing Wandered across the deep: The sailors labouring on their oars Rowed, as in sleep.
All the high pomp of Asia, Charmed by that siren lay, Out of their weary and dreaming minds, Faded away.
Like a bold boy sate their Captain, His glamour withered and gone, In the souls of his brooding mariners, While the song pined on.
Time, like a falling dew, Life, like the scene of a dream, Laid between slumber and slumber, Only did seem....
O Alexander, then, In all us mortals too, Wax thou not bold—too bold On the wave dark-blue!
Come the calm, infinite night, Who then will hear Aught save the singing Of the sea-maids clear?
THE REAWAKENING
Green in light are the hills, and a calm wind flowing Filleth the void with a flood of the fragrance of Spring; Wings in this mansion of life are coming and going, Voices of unseen loveliness carol and sing.
Coloured with buds of delight the boughs are swaying, Beauty walks in the woods, and wherever she rove Flowers from wintry sleep, her enchantment obeying, Stir in the deep of her dream, reawaken to love.
Oh, now begone sullen care—this light is my seeing; I am the palace, and mine are its windows and walls; Daybreak is come, and life from the darkness of being Springs, like a child from the womb, when the lonely one calls.
THE VACANT DAY
As I did walk in meadows green I heard the summer noon resound With call of myriad things unseen That leapt and crept upon the ground.
High overhead the windless air
Throbbed with the homesick coursing cry Of swallows that did everywhere Wake echo in the sky.
Beside me, too, clear waters coursed Which willow branches, lapsing low, Breaking their crystal gliding forced To sing as they did flow.
I listened; and my heart was dumb With praise no language could express; Longing in vain for him to come Who had breathed such blessedness.
On this fair world, wherein we pass So chequered and so brief a stay; And yearned in spirit to learn, alas, What kept him still away.
THE FLIGHT
How do the days press on, and lay Their fallen locks at evening down, Whileas the stars in darkness play And moonbeams weave a crown—
A crown of flower-like light in heaven, Where in the hollow arch of space
Morn's mistress dreams, and the Pleiads seven Stand watch about her place.
Stand watch—O days no number keep Of hours when this dark clay is blind. When the world's clocks are dumb in sleep 'Tis then I seek my kind.
THE TWO HOUSES
In the strange city of Life
Two houses I know well: One wherein Silence a garden hath, And one where Dark doth dwell.
Roof unto roof they stand, Shadowing the dizzied street, Where Vanity flaunts her gilded booths In the noontide glare and heat.
Green-graped upon their walls
An ancient hoary vine Hath clustered their carven, lichenous stones With tendril serpentine.
And ever and anon, Dazed in that clamorous throng, I thirst for the soundless fount that stills Those orchards mute of song.
Knock, knock, nor knock in vain: Heart all thy secrets tell Where Silence a fast-sealed garden hath, Where Dark doth dwell.
FOR ALL THE GRIEF
For all the grief I have given with words May now a few clear flowers blow, In the dust, and the heat, and the silence of birds, Where the lonely go.
For the thing unsaid that heart asked of me Be a dark, cool water calling—calling To the footsore, benighted, solitary, When the shadows are falling.
O, be beauty for all my blindness, A moon in the air where the weary wend, And dews burdened with loving-kindness In the dark of the end.
THE SCRIBE
What lovely things
Thy hand hath made:
The smooth-plumed bird
In its emerald shade, The seed of the grass, The speck of stone
Which the wayfaring ant Stirs—and hastes on!
Though I should sit By some tarn in thy hills, Using its ink
As the spirit wills
To write of Earth's wonders, Its live, willed things, Flit would the ages On soundless wings
Ere unto Z
My pen drew nigh; Leviathan told, And the honey-fly: And still would remain My wit to try—
My worn reeds broken, The dark tarn dry, All words forgotten— Thou, Lord, and I.
FARE WELL
When I lie where shades of darkness
Shall no more assail mine eyes,
Nor the rain make lamentation
When the wind sighs; How will fare the world whose wonder Was the very proof of me?
Memory fades, must the remembered Perishing be?
Oh, when this my dust surrenders
Hand, foot, lip, to dust again, May these loved and loving faces
Please other men!
May the rusting harvest hedgerow
Still the Traveller's Joy entwine, And as happy children gather Posies once mine.
Look thy last on all things lovely, Every hour. Let no night
Seal thy sense in deathly slumber
Till to delight
Thou have paid thy utmost blessing; Since that all things thou wouldst praise Beauty took from those who loved them
In other days.
Printed by T. and A. CONSTABLE, Printers to His Majesty at the Edinburgh University Press
*** END OF THE PROJECT GUTENBERG EBOOK MOTLEY, AND OTHER POEMS ***
Updated editions will replace the previous one—the old editions will be renamed.
Creating the works from print editions not protected by U.S. copyright law means that no one owns a United States copyright in these works, so the Foundation (and you!) can copy and distribute it in the United States without permission and without paying copyright royalties. Special rules, set forth in the General Terms of Use part of this license, apply to copying and distributing Project Gutenberg™ electronic works to protect the PROJECT GUTENBERG™ concept and trademark. Project Gutenberg is a registered trademark, and may not be used if you charge for an eBook, except by following the terms of the trademark license, including paying royalties for use of the Project Gutenberg trademark. If you do not charge anything for copies of this eBook, complying with the trademark license is very easy. You may use this eBook for nearly any purpose such as creation of derivative works, reports, performances and research. Project Gutenberg eBooks may be modified and printed and given away—you may do practically ANYTHING in the United States with eBooks not protected by U.S. copyright law. Redistribution is subject to the trademark license, especially commercial redistribution.
START: FULL LICENSE
THE FULL PROJECT GUTENBERG LICENSE
PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK
To protect the Project Gutenberg™ mission of promoting the free distribution of electronic works, by using or distributing this work (or any other work associated in any way with the phrase “Project Gutenberg”), you agree to comply with all the terms of the Full Project Gutenberg™ License available with this file or online at www.gutenberg.org/license.
Section 1. General Terms of Use and Redistributing Project Gutenberg™ electronic works
1.A. By reading or using any part of this Project Gutenberg™ electronic work, you indicate that you have read, understand, agree to and accept all the terms of this license and intellectual property (trademark/copyright) agreement. If you do not agree to abide by all the terms of this agreement, you must cease using and return or destroy all copies of Project Gutenberg™ electronic works in your possession. If you paid a fee for obtaining a copy of or access to a Project Gutenberg™ electronic work and you do not agree to be bound by the terms of this agreement, you may obtain a refund from the person or entity to whom you paid the fee as set forth in paragraph 1.E.8.
1.B. “Project Gutenberg” is a registered trademark. It may only be used on or associated in any way with an electronic work by people who agree to be bound by the terms of this agreement. There are a few things that you can do with most Project Gutenberg™ electronic works even without complying with the full terms of this agreement. See paragraph 1.C below. There are a lot of things you can do with Project Gutenberg™ electronic works if you follow the terms of this agreement and help preserve free future access to Project Gutenberg™ electronic works. See paragraph 1.E below.
1.C. The Project Gutenberg Literary Archive Foundation (“the Foundation” or PGLAF), owns a compilation copyright in the collection of Project Gutenberg™ electronic works. Nearly all the individual works in the collection are in the public domain in the United States. If an individual work is unprotected by copyright law in the United States and you are located in the United States, we do not claim a right to prevent you from copying, distributing, performing, displaying or creating derivative works based on the work as long as all references to Project Gutenberg are removed. Of course, we hope that you will support the Project Gutenberg™ mission of promoting free access to electronic works by freely sharing Project Gutenberg™ works in compliance with the terms of this agreement for keeping the Project Gutenberg™ name associated with the work. You can easily comply with the terms of this agreement by keeping this work in the same format with its attached full Project Gutenberg™ License when you share it without charge with others.
1.D. The copyright laws of the place where you are located also govern what you can do with this work. Copyright laws in most countries are in a constant state of change. If you are outside the United States, check the laws of your country in addition to the terms of this agreement before downloading, copying, displaying, performing, distributing or creating derivative works based on this work or any other Project Gutenberg™ work. The Foundation makes no representations concerning the copyright status of any work in any country other than the United States.
1.E. Unless you have removed all references to Project Gutenberg:
1.E.1. The following sentence, with active links to, or other immediate access to, the full Project Gutenberg™ License must appear prominently whenever any copy of a Project Gutenberg™ work (any work on which the phrase “Project Gutenberg” appears, or with which the phrase “Project Gutenberg” is associated) is accessed, displayed, performed, viewed, copied or distributed:
This eBook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.
1.E.2. If an individual Project Gutenberg™ electronic work is derived from texts not protected by U.S. copyright law (does not contain a notice indicating that it is posted with permission of the copyright holder), the work can be copied and distributed to anyone in the United States without paying any fees or charges. If you are redistributing or providing access to a work with the phrase “Project Gutenberg” associated with or appearing on the work, you must comply either with the requirements of paragraphs 1.E.1 through 1.E.7 or obtain permission for the use of the work and the Project Gutenberg™ trademark as set forth in paragraphs 1.E.8 or 1.E.9.
1.E.3. If an individual Project Gutenberg™ electronic work is posted with the permission of the copyright holder, your use and distribution must comply with both paragraphs 1.E.1 through 1.E.7 and any additional terms imposed by the copyright holder. Additional terms will be linked to the Project Gutenberg™ License for all works posted with the permission of the copyright holder found at the beginning of this work.
1.E.4. Do not unlink or detach or remove the full Project Gutenberg™ License terms from this work, or any files containing a part of this work or any other work associated with Project Gutenberg™.
1.E.5. Do not copy, display, perform, distribute or redistribute this electronic work, or any part of this electronic work, without prominently displaying the sentence set forth in paragraph 1.E.1 with active links or immediate access to the full terms of the Project Gutenberg™ License.
1.E.6. You may convert to and distribute this work in any binary, compressed, marked up, nonproprietary or proprietary form, including any word processing or hypertext form. However, if you provide access to or distribute copies of a Project Gutenberg™ work in a format other than “Plain Vanilla ASCII” or other format used in the official version posted on the official Project Gutenberg™ website (www.gutenberg.org), you must, at no additional cost, fee or expense to the user, provide a copy, a means of exporting a copy, or a means of obtaining a copy upon request, of the work in its original “Plain Vanilla ASCII” or other form. Any alternate format must include the full Project Gutenberg™ License as specified in paragraph 1.E.1.
1.E.7. Do not charge a fee for access to, viewing, displaying, performing, copying or distributing any Project Gutenberg™ works unless you comply with paragraph 1.E.8 or 1.E.9.
1.E.8. You may charge a reasonable fee for copies of or providing access to or distributing Project Gutenberg™ electronic works provided that:
• You pay a royalty fee of 20% of the gross profits you derive from the use of Project Gutenberg™ works calculated using the method you already use to calculate your applicable taxes. The fee is owed to the owner of the Project Gutenberg™ trademark, but he has agreed to donate royalties under this paragraph to the Project Gutenberg Literary Archive Foundation. Royalty payments must be paid within 60 days following each date on which you prepare (or are legally required to prepare) your periodic tax returns. Royalty payments should be clearly marked as such and sent to the Project Gutenberg Literary Archive Foundation at the address specified in Section 4, “Information about donations to the Project Gutenberg Literary Archive Foundation.”
• You provide a full refund of any money paid by a user who notifies you in writing (or by e-mail) within 30 days of receipt that s/he does not agree to the terms of the full Project Gutenberg™ License. You must require such a user to return or destroy all
copies of the works possessed in a physical medium and discontinue all use of and all access to other copies of Project Gutenberg™ works.
• You provide, in accordance with paragraph 1.F.3, a full refund of any money paid for a work or a replacement copy, if a defect in the electronic work is discovered and reported to you within 90 days of receipt of the work.
• You comply with all other terms of this agreement for free distribution of Project Gutenberg™ works.
1.E.9. If you wish to charge a fee or distribute a Project Gutenberg™ electronic work or group of works on different terms than are set forth in this agreement, you must obtain permission in writing from the Project Gutenberg Literary Archive Foundation, the manager of the Project Gutenberg™ trademark. Contact the Foundation as set forth in Section 3 below.
1.F.
1.F.1. Project Gutenberg volunteers and employees expend considerable effort to identify, do copyright research on, transcribe and proofread works not protected by U.S. copyright law in creating the Project Gutenberg™ collection. Despite these efforts, Project Gutenberg™ electronic works, and the medium on which they may be stored, may contain “Defects,” such as, but not limited to, incomplete, inaccurate or corrupt data, transcription errors, a copyright or other intellectual property infringement, a defective or damaged disk or other medium, a computer virus, or computer codes that damage or cannot be read by your equipment.
1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the “Right of Replacement or Refund” described in paragraph 1.F.3, the Project Gutenberg Literary Archive Foundation, the owner of the Project Gutenberg™ trademark, and any other party distributing a Project Gutenberg™ electronic work under this agreement, disclaim all liability to you for damages, costs and
expenses, including legal fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH DAMAGE.
1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a defect in this electronic work within 90 days of receiving it, you can receive a refund of the money (if any) you paid for it by sending a written explanation to the person you received the work from. If you received the work on a physical medium, you must return the medium with your written explanation. The person or entity that provided you with the defective work may elect to provide a replacement copy in lieu of a refund. If you received the work electronically, the person or entity providing it to you may choose to give you a second opportunity to receive the work electronically in lieu of a refund. If the second copy is also defective, you may demand a refund in writing without further opportunities to fix the problem.
1.F.4. Except for the limited right of replacement or refund set forth in paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE.
1.F.5. Some states do not allow disclaimers of certain implied warranties or the exclusion or limitation of certain types of damages. If any disclaimer or limitation set forth in this agreement violates the law of the state applicable to this agreement, the agreement shall be interpreted to make the maximum disclaimer or limitation permitted by the applicable state law. The invalidity or unenforceability of any provision of this agreement shall not void the remaining provisions.
1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the trademark owner, any agent or employee of the Foundation, anyone providing copies of Project Gutenberg™ electronic works in accordance with this agreement, and any volunteers associated with the production, promotion and distribution of Project Gutenberg™ electronic works, harmless from all liability, costs and expenses, including legal fees, that arise directly or indirectly from any of the following which you do or cause to occur: (a) distribution of this or any Project Gutenberg™ work, (b) alteration, modification, or additions or deletions to any Project Gutenberg™ work, and (c) any Defect you cause.
Section 2. Information about the Mission of Project Gutenberg™
Project Gutenberg™ is synonymous with the free distribution of electronic works in formats readable by the widest variety of computers including obsolete, old, middle-aged and new computers. It exists because of the efforts of hundreds of volunteers and donations from people in all walks of life.
Volunteers and financial support to provide volunteers with the assistance they need are critical to reaching Project Gutenberg™’s goals and ensuring that the Project Gutenberg™ collection will remain freely available for generations to come. In 2001, the Project Gutenberg Literary Archive Foundation was created to provide a secure and permanent future for Project Gutenberg™ and future generations. To learn more about the Project Gutenberg Literary Archive Foundation and how your efforts and donations can help, see Sections 3 and 4 and the Foundation information page at www.gutenberg.org.
Section 3. Information about the Project Gutenberg Literary Archive Foundation
The Project Gutenberg Literary Archive Foundation is a non-profit 501(c)(3) educational corporation organized under the laws of the state of Mississippi and granted tax exempt status by the Internal Revenue Service. The Foundation’s EIN or federal tax identification number is 64-6221541. Contributions to the Project Gutenberg Literary Archive Foundation are tax deductible to the full extent permitted by U.S. federal laws and your state’s laws.
The Foundation’s business office is located at 809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up to date contact information can be found at the Foundation’s website and official page at www.gutenberg.org/contact
Section 4. Information about Donations to the Project Gutenberg Literary Archive Foundation
Project Gutenberg™ depends upon and cannot survive without widespread public support and donations to carry out its mission of increasing the number of public domain and licensed works that can be freely distributed in machine-readable form accessible by the widest array of equipment including outdated equipment. Many small donations ($1 to $5,000) are particularly important to maintaining tax exempt status with the IRS.
The Foundation is committed to complying with the laws regulating charities and charitable donations in all 50 states of the United States. Compliance requirements are not uniform and it takes a considerable effort, much paperwork and many fees to meet and keep up with these requirements. We do not solicit donations in locations where we have not received written confirmation of compliance. To SEND DONATIONS or determine the status of compliance for any particular state visit www.gutenberg.org/donate.
While we cannot and do not solicit contributions from states where we have not met the solicitation requirements, we know of no
prohibition against accepting unsolicited donations from donors in such states who approach us with offers to donate.
International donations are gratefully accepted, but we cannot make any statements concerning tax treatment of donations received from outside the United States. U.S. laws alone swamp our small staff.
Please check the Project Gutenberg web pages for current donation methods and addresses. Donations are accepted in a number of other ways including checks, online payments and credit card donations. To donate, please visit: www.gutenberg.org/donate.
Section 5. General Information About Project Gutenberg™ electronic works
Professor Michael S. Hart was the originator of the Project Gutenberg™ concept of a library of electronic works that could be freely shared with anyone. For forty years, he produced and distributed Project Gutenberg™ eBooks with only a loose network of volunteer support.
Project Gutenberg™ eBooks are often created from several printed editions, all of which are confirmed as not protected by copyright in the U.S. unless a copyright notice is included. Thus, we do not necessarily keep eBooks in compliance with any particular paper edition.
Most people start at our website which has the main PG search facility: www.gutenberg.org.
This website includes information about Project Gutenberg™, including how to make donations to the Project Gutenberg Literary Archive Foundation, how to help produce our new eBooks, and how to subscribe to our email newsletter to hear about new eBooks.
