Algebra: From the Viewpoint of Galois Theory

369 Pages·2018·4.46 MB·English

by  Siegfried Bosch

Birkhäuser Advanced Texts
Basler Lehrbücher
Siegfried Bosch
Algebra
From the Viewpoint of Galois Theory

Birkha¨userAdvancedTextsBaslerLehrbu¨cher
Serieseditors
StevenG.Krantz,WashingtonUniversity,St.Louis,USA
ShrawanKumar,UniversityofNorthCarolinaatChapelHill,ChapelHill,USA
JanNekováˇr,UniversitéPierreetMarieCurie,Paris,France
Moreinformationaboutthisseriesat http://www.springer.com/series/4842
Siegfried Bosch
Algebra
From the Viewpoint of Galois Theory
Siegfried Bosch
Mathematisches Institut
Westfälische Wilhelms-Universität
Münster, Germany
ISSN 1019-6242 ISSN 2296-4894 (electronic)
Birkhäuser Advanced Texts Basler Lehrbücher
ISBN 978-3-319-95176-8 ISBN 978-3-319-95177-5 (eBook)
https://doi.org/10.1007/978-3-319-95177-5
Library of Congress Control Number: 2018950547
Mathematics Subject Classification (2010): 12-01, 13-01, 14-01
Translation from the German language edition: Algebra by Siegfried Bosch, Copyright © Springer-Verlag
GmbH Deutschland, 2013. All Rights Reserved. ISBN 978-3-642-39566-6

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Preface
The material presented here can be divided into two parts. The first, some-
times referred to as abstract algebra, is concerned with the general theory of
algebraic objects such as groups, rings, and fields, hence, with topics that are
also basic for a number of other domains in mathematics. The second centers
around Galois theory and its applications. Historically, this theory originated
fromtheproblem ofstudying algebraicequations, aproblem that,aftervarious
unsuccessful attempts to determine solution formulas in higher degrees, found
itscompleteclarificationthroughthebrilliantideasofE.Galois.ToconvertGa-
lois’s approach into a comprehensible theory, in other words, to set up Galois
theory, hastaken quiteaperiodoftime. Thereasonisthatsimultaneously sev-
eral new concepts ofalgebrawere emerging and had tobedeveloped asnatural
prerequisites. In fact, the study of algebraic equations has served as a motivat-
ing terrain for a large part of abstract algebra, and according to this, algebraic
equations will be visible as a guiding thread throughout the book.
To underline this point, I have included at the beginning a historical in-
troduction to the problem of solving algebraic equations. Later, every chapter
begins with some introductory remarks on “Background and Overview,” where
I give motivation for the material that follows and where I discuss some of its
highlightsonaninformallevel. Incontrasttothis,theremaining “regular”sec-
tions (some of them optional, indicated by a star) go step by step, elaborating
thecorrespondingsubjectinfullmathematicalstrength.Ihavetriedtoproceed
in a way as simple and as clear as possible, basing arguments always on “true
reasons,”inotherwords, withoutresortingtosimplifying adhocsolutions. The
textshouldthereforebeusefulfor“any”courseonthesubjectandevenforself-
study, certainly since it is essentially self-contained, up to a few prerequisites
fromlinearalgebra.Each sectionendswithalist ofspeciallyadaptedexercises,
some of them printed in italics to signify that there are solution proposals in
the appendix.
Onmanyoccasions,Ihavegivencoursesonthesubjectofthisbook,usually
in units of two for consecutive semesters. In such courses I have addressed the
“standardprogram”consistingoftheunstarredsections.Thelatteryieldawell-
founded and direct access to the world of algebraic field extensions, with the
fundamental theorem of Galois theory as a first milestone. Also let me point
out that group theory has been split up into an elementary part in Chapter 1
andamoreadvancedpartlaterinChapter5thatisneededfortheapplications
V
VI Preface
of Galoistheory. Of course, if preferred, Chapter 5 can be covered immediately
afterChapter1.Finally,theoptionalstarredsectionscomplement thestandard
program or, in some cases, provide a first view on nearby areas that are more
advanced. Such sections are particularly well suited for seminars.
Thefirstversions ofthisbookappearedinGermanashandoutsformystu-
dents. They were later compiled into a book on algebra that appeared in 1993.
I’m deeply indebted tomy students and colleaguesfortheir valuablecomments
and suggestions. All this found its way into later editions. The present English
editionisatranslationandcriticalrevisionoftheeighthGermaneditionof2013.
Here my thanks go to my colleague and friend Alan Huckleberry, with whom I
discussed several issues of the English translation, as well as to Birkh¨auser and
its editorial team for the smooth editing and publishing procedure.
Mu¨nster, May 2018 Siegfried Bosch
Contents
Introduction: On the Problem of Solving Algebraic Equations . . . . . . 1
1 Elementary Group Theory . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Cosets, Normal Subgroups, Factor Groups . . . . . . . . . . . . 15
1.3 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Rings and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Polynomial Rings in One Variable . . . . . . . . . . . . . . . . 26
2.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Ring Homomorphisms, Factor Rings . . . . . . . . . . . . . . . 35
2.4 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 Polynomial Rings in Several Variables . . . . . . . . . . . . . . 51
2.6 Zeros of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 57
2.7 A Theorem of Gauss . . . . . . . . . . . . . . . . . . . . . . . . 59
2.8 Criteria for Irreducibility . . . . . . . . . . . . . . . . . . . . . 65
2.9 Theory of Elementary Divisors* . . . . . . . . . . . . . . . . . . 67
3 Algebraic Field Extensions . . . . . . . . . . . . . . . . . . . . . . . 83
3.1 The Characteristic of a Field . . . . . . . . . . . . . . . . . . . 85
3.2 Finite and Algebraic Field Extensions . . . . . . . . . . . . . . 87
3.3 Integral Ring Extensions* . . . . . . . . . . . . . . . . . . . . . 94
3.4 Algebraic Closure . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.6 Separable Field Extensions . . . . . . . . . . . . . . . . . . . . 111
3.7 Purely Inseparable Field Extensions . . . . . . . . . . . . . . . 119
3.8 Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.9 Beginnings of Algebraic Geometry* . . . . . . . . . . . . . . . . 126
4 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.1 Galois Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.2 Profinite Galois Groups* . . . . . . . . . . . . . . . . . . . . . 142
4.3 The Galois Group of an Equation . . . . . . . . . . . . . . . . . 153
4.4 Symmetric Polynomials, Discriminant, Resultant* . . . . . . . 162
4.5 Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
VII
VIII Contents
4.6 Linear Independence of Characters . . . . . . . . . . . . . . . . 186
4.7 Norm and Trace . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.8 Cyclic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.9 Multiplicative Kummer Theory* . . . . . . . . . . . . . . . . . . 200
4.10 General Kummer Theory and Witt Vectors* . . . . . . . . . . . 205
4.11 Galois Descent* . . . . . . . . . . . . . . . . . . . . . . . . . . 224
5 More Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
5.1 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
5.2 Sylow Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5.3 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . 245
5.4 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6 Applications of Galois Theory . . . . . . . . . . . . . . . . . . . . . 255
6.1 Solvability of Algebraic Equations . . . . . . . . . . . . . . . . 256
6.2 Algebraic Equations of Degree 3 and 4* . . . . . . . . . . . . . . 264
6.3 Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . 272
6.4 Compass and Straightedge Construction . . . . . . . . . . . . . 275
7 Transcendental Field Extensions . . . . . . . . . . . . . . . . . . . . 283
7.1 Transcendence Bases . . . . . . . . . . . . . . . . . . . . . . . . 284
7.2 Tensor Products* . . . . . . . . . . . . . . . . . . . . . . . . . . 290
7.3 Separable, Primary, and Regular Extensions* . . . . . . . . . . 301
7.4 Differential Calculus* . . . . . . . . . . . . . . . . . . . . . . . 311
Appendix: Solutions to Exercises . . . . . . . . . . . . . . . . . . . . . . 323
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Glossary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Introduction
On the Problem of Solving Algebraic Equations
The word algebra is of Arabic origin (ninth century AD) and means doing
calculations on equations, such ascombining different termsof theequation, or
changingtermsbysuitablemanipulationsonbothsidesoftheequation.Herean
equationismeantasarelationbetween known quantities, so-calledcoefficients,
and unknown quantities or variables, whose possible value is to be determined
by means of the equation. In algebra one is mostly interested in polynomial
equations, for example of type
2x3+3x2+7x−10=0,
where x stands for the unknown quantity. Such an equation will be referred to
asanalgebraic equationforx.Itsdegree isgivenbytheexponentofthehighest
power ofxthatactuallyoccursintheequation.Algebraicequationsofdegree1
are called linear. The study of these or, more generally, of systems of linear
equations in finitely many variables, is a central problem in linear algebra.
On the other hand, algebra in the sense of the present book is about alge-
braicequationsofhigher degreein onevariable. Intoday’slanguage, thisisthe
theoryoffieldextensionstogetherwithallitsabstractconcepts,includingthose
ofgroup-theoreticnaturethat,intheircombination,makepossibleaconvenient
and comprehensive treatment of algebraic equations. Indeed, even on an “ele-
mentary” level, modern algebra is much more influenced by abstract methods
and concepts than one is used to from other areas, for example from analy-
sis. The reason becomes apparent if we follow the problem of solving algebraic
equations from a historical viewpoint, as we will briefly do in the following.
In the beginning, algebraic equations were used essentially in a practical
manner, to solve certain numerical “exercises.” For example, a renowned prob-
lem of ancient Greece (c. 600 BC – 200 AD) is the problem on the duplication
ofthecube. Given acubeofedgelength1,itasks todetermine theedgelength
ofacubeofdoublevolume.Inotherwords,theproblemistosolvethealgebraic
equation√x3 = 2, which is of d√egree 3. Today the solution would be described
by x= 32. However, what is 32 if only rational numbers are known? Since it
was not possible to find a rational number whose third power is 2, one had to
content ones√elf with approximate solutions and hence sufficiently good approx-
imations of 32. On the other hand, the duplication of the cube is a problem of
1