Table Of ContentJuliusz 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
,
