Juliusz Brzezin´ski Galois Theory through Exercises 123 JuliuszBrzezin´ski DepartmentofMathematicalSciences UniversityofGothenburg Sweden ChalmersUniversityofTechnology Sweden ISSN1615-2085 ISSN2197-4144 (electronic) SpringerUndergraduateMathematicsSeries ISBN978-3-319-72325-9 ISBN978-3-319-72326-6 (eBook) https://doi.org/10.1007/978-3-319-72326-6 LibraryofCongressControlNumber:2017964367 ©SpringerinternationalPublishingAG,partofSpringerNature2018 Preface ThisBook ThepresentbookevolvedfromdifferentcollectionsofexerciseswhichIwrotewhen I started my teaching duties at the beginning of the 1960s, first at the University of Warsaw, and 10 years later, at the University of Gothenburgand the Chalmers UniversityofTechnology.A fewtextbooksavailableatthetime,like EmilArtin’s seminal book “Galois Theory”, which was used frequently,or van der Waerden’s “Algebra”(initspartconcernedwithGaloistheory)neededsupplementarymaterial intheformofsimpleexamplesandsomeadvancedexercisesforinterestedstudents. Usually, a course in Galois theory followed one or two courses in algebra, which could present some necessary previous knowledge. A common experience of all the courses were the reactions of my students, complaining that it is usually relatively easy to start with a standard exercise in calculus, while an exercise in algebra presents no clues as to how to start solving it, even if you know the relevant theorems. Naturally, the degree of difficulty of the exercises can create an obstacle in any area of mathematics, but in algebra such opinions are often relatedtorathermundaneproblems.Therefore,thisphenomenondeservesreflection and explanation. There are probably several reasons for such a perception of the subject. One possibility is the characterof the tasks related to the same objectsin differentcontexts.Forexample,incalculus,concretenumbersorfunctionswillhave alreadybeen encounteredin high schooland,usually, the first exercisesare about concrete objects of this kind. Similar objects appear in algebra, but it is rather a generalpropertyofawholesetofnumbersorfunctionswhichisrelevant.Another essentialdifferenceisthepresenceofseveralnewnotionsrelatedtothesets,which are not necessarily well known from previous encounters with mathematics, e.g. whatarethepropertiesofadditionormultiplicationofdifferenttypesofnumbers. Still another aspect is probably the character of the proofs and the possibility of transferring the arguments from them to concrete situations. The proofs of the theoremsin calculus usually give a clue of how to handle concrete objects, while in algebra a proof of a general property of a structure satisfying some conditions usually says very little about suitable examples of such structures and even less about the methods one can use to determine whether or not a concrete structure has a particular property. This is one of the reasons why the adjective “abstract” is used in the names of some algebra courses. Perhaps the only way to address these difficulties, which may appear in connection with courses in algebra, is to exposethestudentstomanyconcreteexamplesandtogivethemthechancetosolve manyexercises.ThisistheprimarypurposeofthistextasabookonGaloistheory motivatedbyexercisesandexamplesoftheirsolutions. AFew Words onthe Subject Galoistheorygrewfromthedesiretosolveaveryconcretemathematicalproblem related to polynomial equations of degree higher than 4. Already during the 18th century, some mathematicians suspected that the efforts to find general formulae for solutions of quintics (polynomial equations of degree 5) were futile in spite of the existence of such formulae for equations of degree lower than 5. Galois proved that there are quintics with rational coefficients whose zeros cannot be expressed by means of these coefficients using the four arithmetical operations (addition,subtraction,multiplication,division)andextractingroots.Asimilarresult concerningso-calledgeneralpolynomialequationsofdegree5(polynomialswhose coefficientsarevariables)waspublished(butnotexactlyproved)byRuffiniin1799 andfinallyprovedbyAbelindependentlyofGalois.Theunsolvabilityofthequintic isaveryconcrete,interestinganddeepproblem,whichprovidesverygoodreasons to go into the details of a theory which gives its solution. The problem is easy to explainandcreatesmotivationforthestudentstousealotoftheoryfromprevious courses in algebra as well as to see the need for a suitable language in order to formulateitinmathematicalterms. However, learning Galois theory today is not only motivated by this result concerningpolynomialequations.ThetheorydevelopedbyGaloishastremendous value as a path to modern mathematics. In order to solve a concrete algebraic problem,Galoistranslatedittoanew(abstract)languageofgroupsandfields.This was not only a great contribution to these theories (e.g. Galois introduced many notionsingrouptheory),butalsogavelaterideasaboutthepossibilityofusingnew mathematicalstructureslikegroups,rings,fields,linearspaces,topologicalspaces and many other (abstract) structures in order to reformulate and solve concrete mathematicalproblems.Thus,Galoistheorywasasourceoffuturedevelopmentsof newtheorieshavingstillbroaderapplicability.Galoistheoryitselfalsofoundmany important extensions, generalizations and applications, for example, in number theory,topologyandthetheoryofgrouprepresentations. TheStructure oftheBook Since it is impossible to exactly define a “standard selection” of notions and theoremswhichconstituteacourseinGaloistheory,thetextisstructuredinsucha waythatastudentwillbeabletouseitasanexercisebookevenwithoutprevious knowledge of the subject. Chapters 1–15 together with Chap.17 give more than a very standard course could contain. In the following section, we describe the contents of the book in a way which explains how it may be used as a textbook. Thebookwasoftenusedasaprimarysourceofknowledgeand,inseveralcases,by thosestudentswhostudiedGaloistheoryontheirownwithlimitedassistancefrom theinstructor. Everychapter,outofthe first15,starts witha presentationofthemainnotions and theorems followed by a comprehensive set of exercises which explain the fundamental facts about them in a practical way. The exercises graduate from simple, often numerical, to more complexand theoretical. At the same time, they are usually ordered from more standard to more special. As a rule, the exercises giveninChaps.1–15shouldbeconsideredaspartofabasiccourseinthesubject. Chapter 16 contains a choice of “one hundred” supplementary problems without hintsorsolutions. Incontrast,Chap.18includessomelimitedhintsandanswers(whenananswer canbegiven)totheexercisesinChaps.1–15.Asusual,andinparticularinalgebraic problems,itis quite oftenpossible to givedifferentsolutions.Therefore,the hints cannot be considered as thoughts in “Mao’s Little Red Book”. This is even more importantinChap.19,whichcontainsanumberofstandardexamplesandcomplete solutions of more difficult exercises. Let us note that the standard exercises from Chaps.1to15(inparticular,thoseatthebeginningofeachchapter)arenotsolved in the book—almost every such exercise is illustrated in Chap.19 by an example of a similar nature in order to show how to handle standard problems (often of a computational nature) and to create the possibility of using the exercises in Chaps.1–15 as homework. Reading the examples with solutions is a part of the learning process and is usually an ingredient in any textbook. On the other hand, completesolutionsofsomemorespecialproblems,aspresentedinChap.19,haveat leastthreedifferentfunctions.Firstofall,theycontainanumberofusefulauxiliary results, which are usually proved in the main texts of more standard textbooks. Then, some solutions are examples of how to work with the notions and handle similarproblems(thereisarichchoiceofproblemswithoutsolutionsinChap.16). Finally,someofthesolutionspresentedinthischaptermayberegardedasthelast resort,whenseriousattemptstosolveaproblemwerefruitless,orperhapsinorder tocompareone’sownsolutiontoadifferentonesuggestedinthebook. Chapter17containsdetailedproofsofallthetheoremsusedinChaps.1–15.Each ofthese chaptersstarts withan introductorypartin whichallrelevantnotionsand theoremsareformulatedanddiscussed.Thepresentationoftheproofsinaseparate chapterisaconvenienttechnicalsolution.Inparticular,itmakesthepresentationof thetheoreticalbackgroundof eachchaptermoretransparentandclear.Italso lays stressontheimportanceofexamplesandexercisesinthelearningprocess.However, theproofsshouldbestudiedindirectconnectionwitheachchapter,inparticular,in thosecaseswhentheproofcontainscluestopracticalconstructionsofthoseobjects whoseexistenceisstatedinthetheorem.Afewofthetheorems,whichareusually a part of previouscourses in algebra, are included in the Appendix. Some others, whoseproofscanbeconsideredasanapplicationofGaloistheory,areincludedin theexercises.Theirproofscanbeconsideredasoptional. Manychaptersincludeasectionon“usingcomputers”,whichgivesexamplesof how to apply the software package Maple to concrete problems in Galois theory. Maple has a very extensive library of algebraic functions, which helps to define fundamentalalgebraic structures and carry out different computations. Using this packagecreatesmanypossibilitiestosupplementalotofexercises,whichcontribute toa betterunderstandingandareimpossiblewithouttheassistance ofa computer. Thereareothercomputerpackageswhichcanbeused,likePARI/GPorSAGE,but thechoiceofMapleisduetotheeaseoftheinterface. TheAppendixattheendofthetextisrelativelylong,andcontains,inprinciple, all the necessary material corresponding to a first course in algebra, discussing groups, rings and fields. The purpose is to provide a convenientsource for direct references to the background material necessary in any text on Galois theory. However, even if we give some proofs, there are only a few examples and no exercises, so the purpose is rather to recall the necessary definitions and some fundamental results in the form of a glossary for the convenience of the Reader whohasforgottenpartsofsuchacourseandneedsashortreminder.TheAppendix also contains a few topics which are essential in this book, but are not a part of standardcoursesongroups,ringsandfields. Adviceto theReader Westartherewithaveryinformal“introductiontoGaloistheory”inordertogive some of the general ideas behind it. We want to use this as a guide to the text of the book, showing how to follow the book and how to choose among different directions. Consider the polynomial equation f.X/ D X4 (cid:2) 10X2 C 1 D 0. It has four p p soplutions x1;2;3;4 Dp˙ 2˙ 3 (note that f.X/ D .X2 C1/2 (cid:2)12X2 D .X2 (cid:2) 2 3XC1/.X2C2 3XC1/).Thesolutionsdefineafieldextensionoftherational numbersQ(herethecoefficientsoff.X/arerational).Thisextensionisthesmallest setofnumberswhichcontainsboththefieldofthecoefficientsQandthesolutions p p oftheequationf.X/ D 0.ItisnotdifficulttocheckthatitisthefieldQ. 2; 3/. Thisfieldis calleda splittingfieldoff.X/.Thefirstobservationisthatbeingable to express a solution in terms of the numbers from the field of the coefficients of f.X/ using the four arithmetical operations (addition, subtraction, multiplication, division)andextractingrootsmeansthatthereisaformulaforasolution(e.g.x1 D p p 2C 3).Inthiscase,wehavesuchaformula,sincethesplittingfieldisobtained from the rational numbers by adjoining roots of numbers belonging to the field containingthecoefficients.Thisprocessofbuilding“aformula”maybemuchmore involved(itispossiblethatinaformulatherearerootsofotherroots)butitisrather clear thatthe existenceofan algebraicformulaexpressinga solutionby meansof thefourarithmeticaloperations(addition,subtraction,multiplication,division)and extractingrootsmaybeformulatedintermsofsuitablefieldextensions(suchfield extensionsarecalledradical). Galoisstudiedallpermutationsofthesolutionswhichdonotchangeallrational relations among them. Rational relations are different polynomial equalities in x p p i (piD 1;p2;3;4)withprationpalcoefficienpts,whpicharetrue.Letx1 D 2C 3,x2 D 2(cid:2) 3,x3 D (cid:2) 2C 3,x4 D (cid:2) 2(cid:2) 3.Asexamplesofsuchrelations,we havex1 Cx4 D 0,x1x2 D (cid:2)1, andsimilarly,x2Cx3 D 0, x3x4 D (cid:2)1.Thereare 24permutationsofx1;x2;x3;x4.TheyformthegroupS4ofallpermutationsoffour differentsymbols, here, the numbers x. But in order to simplify the notation, we i will replace a permutationof x1;x2;x3;x4 by a suitable permutationof the indices 1;2;3;4.Forexample,thepermutationx2;x1;x4;x3willbedenotedby (cid:2) (cid:3) 1 23 4 2 14 3 or, shortly, .1;2/.3;4/. The symbol .1;2/ denotes the permutation replacing x1;x2;x3;x4 by x2;x1;x3;x4. So let us take only those permutations which do not changethesetofrationalrelationswhichwehavefound.Itisratherevidentthatthe permutationsofx1;x2;x3;x4,whichdonotchangethesetofrelationsx1Cx4 D0, x1x2 D(cid:2)1,x2Cx3 D0,x3x4 D(cid:2)1are.1;2/.3;4/,.1;3/.2;4/,.1;4/.2;3/andof course,theidentitypermutation,whichtakeseachsolutionsontoitself.Theidentity is usually denotedby .1/. If we take any other of the 20 remainingpermutations, for example, .1;2;3;4/, which means that x1;x2;x3;x4 goes to x2;x3;x4;x1, then, forexample,therelationx1Cx4 D 0goestox2Cx1 D 0,whichisnottrue.The permutation.1;2/replacesx1 Cx4 D 0 byx2 Cx4 D 0 whichis nottrueand so on.Infact,ashortcheckshowsthatthefourpermutationswhichwehavefoundare exactlythosewhichtakethesetofthefourrelationsintoitself.Theyformagroup withfourelements:.1/,.1;2/.3;4/,.1;3/.2;4/,.1;4/.2;3/.ThisisjusttheGalois groupoftheequationX4(cid:2)10X2C1D0. Galois related a finite group to each polynomial equation f.X/ D 0 with coefficients in an arbitrary field (here the coefficients are in the field of rational numbers Q). Any polynomial defines a field over the field of its coefficients generated by all its zeros (in a field which contains all zeros like the complex numbers when the field of the coefficients is the field of rational numbers). This field, called a splitting field of the polynomial over its field of the coefficients, containsjustallinformationabouttherelationsamongthesolutionsoftheequation f.X/ D 0. In the case of the polynomialf.X/ D X4 (cid:2)10X2 C1, such a field is p p Q. 2; 3/.AsplittingfielddefinestheGaloisgroupofthepolynomial.Thenotion of a splitting field and its groupmakesthe study of the allowedpermutations(the elementsoftheGaloisgroup)muchmoreprecisethantheprocessdescribedinthe introductoryexample.Galoisestablishedacorrespondencebetweenthesubgroups oftheGaloisgroupofapolynomialandthesubfieldsofitssplittingfield.Usingthis correspondence,theexistenceofaformulaforasolutionofanequationwasrelated toaverydistinctivefeatureoftheGaloisgroup,which,infact,mostoftheGalois groupsdonothave(thegroupissolvable—apropertyintroducedbyGalois).Inthat way,itispossibletogiveexamplesofpolynomialequationswhosesolutionscannot be expressed by means of the four arithmetical operations (addition, subtraction, multiplication,division)andextractingroots—itsufficestofindapolynomialwhose Galoisgroupislackingthedistinctivepropertyofsolvablegroups. Now we are in a positionto describethe contentsof the bookand the possible waysthroughthetopicsdiscussedindifferentchapters.Theirinterdependenciesare showninthegraphbelow. As the unsolvabilityof some quinticsand higherdegreeequationsis chosenin thistextasthemainconcretemotivatingproblem,wediscusspolynomialequations in Chap.1. It explains some methods used for solving polynomial equations of degree lower than 5 and gives a few historical facts concerning them. It is not necessary to pay too much attention to this chapter, which in mathematical sense is loosely related to Chap.3. Moreover,many computer packages using symbolic algebra make it possible to obtain exact solutions of polynomial equations, when it is possible (at least up to degree 4). But the purpose of this chapter is rather to describethe historicaldevelopmentof Galoistheoryandto givea feelingofwhat shouldbeexpectedfromageneralformulaforthesolutionofanalgebraicequation. The main contents, covering a standard course in Galois theory, is presented in Chaps.2–9. In Chap.2, we fix the terminologyconcerningfield extensionsand discuss the notion of the characteristic of a field, which usually does not get so muchattentioninintroductorycoursesinalgebra.Chapter3isaboutpolynomials. Essentially, we repeat several known facts aboutpolynomials, but we concentrate onthenotionofirreducibility,whichisimportantwhenweconstructsplittingfields andwanttoinvestigatetheirproperties.Chapter4treatsfundamentalpropertiesof algebraicfieldextensions.InChap.5,weintroducethenotionofthesplittingfields ofpolynomialsandinvestigatetheirmostimportantproperties.InChap.6,wedefine andstudyautomorphismgroupsoffieldextensions.WecallthemGaloisgroupsin ordertosimplifytheterminologyandformulationsofexercises(but,sometimesin theliterature,thenotionofGaloisgroupisrestrictedtoGaloisextensionsoffields). A Galois extension is an extension which has two properties—it is normal and separable. These two propertiesare introducedand explainedin Chaps.7 (normal extensions) and 8 (separable extensions). Chapter 9 contains the main theorems of Galois theory. One of these theorems describes a correspondence between the subgroupsoftheGaloisgroupofapolynomial(overitsfieldofcoefficients)andthe subfields of its splitting field (over the same field). This “Galois correspondence” is one of the best known theoretical results of Galois theory, which has many generalizationsandcounterpartsinmanyothermathematicalsituations. AfterthepathstartingwithChap.2andendingwithChap.9,therearedifferent waystocontinue.Theshortestpathtothesolutionofthemotivatingproblemgoes toChap.13,wherethereisa proofoftheunsolvabilityofgeneralquinticsandthe general equations of degree higher than 4. Chapter 13 needs some knowledge of solvablegroupsdiscussedinChap.12(thischapterisonlyconcernedwithgroups). Notice that in Chap.13, we use Galois theory in order to give one of the proofs thatthefieldofcomplexnumbersisalgebraicallyclosed(seeExercise13.14)—this resultisusuallycalledthefundamentaltheoremofalgebra(seeChap.1).Anatural supplement to Chap.13 is Chap.15, which presents some practical methods for findingGaloisgroupsofconcretepolynomialsaswellasconstructingpolynomials withgivenGaloisgroups.Thetheoremsonresolventsprovedinthischapterareused incomputerimplementationsofalgorithmsforcomputationsofGaloisgroups.We also use them in a proofof a theoremof Richard DedekindT.15.4,which givesa way to constructinteger polynomialswith the largestpossible Galois groupwhen thedegreeisfixed.Letusnotethatthereisaveryquickroutetotheunsolvability ofquinticsusingNagell’sidea(seeExercise13.6). Another path from Chap.9 goes to Chap.10 and, possibly, to Chap.11. In Chap.10,westudycyclotomicfields.Theseextensionsoftherationalnumbersby a root of unity are very good examples of how Galois theory works in practice. Atthesametime,thecyclotomicfieldsareveryinterestingandimportantfortheir applicationsinnumbertheory.Chapter10hasanaturalextensioninChap.11,which is the most “theoretical”partof the text. Here we study severalinteresting results which are related to different applications and extensions of Galois theory. The normalbasistheorem(NBT)isaboutbasesofGaloisextensions,whicharenaturally related to the Galois groupsand can be best formulatedin terms of modulesover groups. Hilbert’s Theorem 90 is a famous result with many applications—one of them is a description of all cyclic Galois field extensions (over fields containing suitable roots of unity). Kummer’s extensions, which are also discussed in this chapter,generalizeGalois extensionswith cyclic groupsto arbitraryfinite abelian groups(with similar assumptionsaboutthe presence of rootsof unity).As one of theapplicationsinChap.11,wegiveafulldescriptionofcubicandquarticGalois extensionsovertherationalnumbers. Finally, the third path from Chap.9 goes to Chap.14, in which Galois theory finds applications to classical problems concerning geometric straightedge-and- compass constructions. The truth is that most of this chapter could already be discussed afterChap.4 with onlya verymodestknowledgeof field extensions.In fact, geometric constructionsare often discussed as an applicationin introductory courses in algebra. But there are some results for which a deeper knowledge of Galois theory is very helpful, in particular, when one wants to prove that some geometricconstructionsarepossible(forexample,inGauss’stheoremaboutregular polygons). Mölnlycke,Sweden JuliuszBrzezin´ski January,2018 Interrelations ofChaps.1–15 ,