Algebra 2: Linear Algebra, Galois Theory, Representation theory, Group extensions and Schur Multiplier

440 Pages·2017·4.36 MB·English

by  Ramji Lal (auth.)

Infosys Science Foundation Series in Mathematical Sciences
Ramji Lal
Algebra 2
Linear Algebra, Galois Theory,
Representation Theory, Group
Extensions and Schur Multiplier
Infosys Science Foundation Series
Infosys Science Foundation Series in Mathematical
Sciences
Series editors
Gopal Prasad, University of Michigan, USA
Irene Fonseca, Mellon College of Science, USA
Editorial Board
Chandrasekhar Khare, University of California, USA
Mahan Mj, Tata Institute of Fundamental Research, Mumbai, India
Manindra Agrawal, Indian Institute of Technology Kanpur, India
S.R.S. Varadhan, Courant Institute of Mathematical Sciences, USA
Weinan E, Princeton University, USA
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Ramji Lal
Algebra 2
Linear Algebra, Galois Theory,
Representation Theory, Group Extensions
and Schur Multiplier
123
Ramji Lal
HarishChandra Research Institute (HRI)
Allahabad, Uttar Pradesh
India
ISSN 2363-6149 ISSN 2363-6157 (electronic)
Infosys Science FoundationSeries
ISSN 2364-4036 ISSN 2364-4044 (electronic)
Infosys Science FoundationSeries in MathematicalSciences
ISBN978-981-10-4255-3 ISBN978-981-10-4256-0 (eBook)
DOI 10.1007/978-981-10-4256-0
LibraryofCongressControlNumber:2017935547
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Dedicated to the memory of
my mother
(Late) Smt Murti Devi,
my father
(Late) Sri Sankatha Prasad Lal, and
my father like brother
(Late) Sri Gopal Lal
Preface
Algebra has played a central and decisive role in all branches of mathematics and,
inturn,inallbranchesofscienceandengineering.Itisnotpossibleforalecturerto
cover, physically in a classroom, the amount of algebra which a graduate student
(irrespective of the branch of science, engineering, or mathematics in which he
preferstospecialize)needstomaster.Inaddition,thereareavarietyofstudentsina
class. Some of them grasp the material very fast and do not need much of assis-
tance. At the same time, there are serious students who can do equally well by
putting a little more effort. They need some more illustrations and also more
exercisestodeveloptheirskillandconfidenceinthesubjectbysolvingproblemson
theirown.Again,itisnotpossibleforalecturertodosufficientlymanyillustrations
and exercises in the classroom for the aforesaid purpose. This is one of the con-
siderations which prompted me to write a series of three volumes on the subject
starting from the undergraduate level to the advance postgraduate level. Each
volume is sufficiently rich with illustrations and examples together with numerous
exercises. These volumes also cater for the need of the talented students with
difficult, challenging, and motivating exercises which were responsible for the
furtherdevelopmentsinmathematics.Occasionally,theexercisesdemonstratingthe
applications in different disciplines are also included. The books may also act as a
guidetoteachersgivingthecourses.Theresearchers workinginthefieldmayalso
find it useful.
The first volume consists of 11 chapters, which starts with language of mathe-
matics(logicandsettheory)andcentersaround theintroductiontobasicalgebraic
structures, viz., groups, rings, polynomial rings, and fields together with funda-
mentalsinarithmetic.Thisvolumeservesasabasictextforthefirst-yearcoursein
algebra at the undergraduate level. Since this is the first introduction to the
abstract-algebraic structures, we proceed rather leisurely in this volume as com-
pared with the other volumes.
The present (second) volume contains 10 chapters which includes the funda-
mentals of linear algebra, structure theory of fields and the Galois theory, repre-
sentationtheoryofgroups,andthetheoryofgroupextensions.Itisneedlesstosay
that linear algebra is the most applicable branch of mathematics, and it is essential
vii
viii Preface
for students of any discipline to develop expertise in the same. As such, linear
algebraisanintegralpartofthesyllabusattheundergraduatelevel.Indeed,avery
significant andessentialpart (Chaps. 1–5)of linear algebracovered inthis volume
doesnotrequireanybackgroundmaterialfromVolume1ofthebookexceptsome
amount of set theory. General linear algebra over rings, Galois theory, represen-
tation theory of groups, and the theory of group extensions follow linear algebra,
andindeedthesearepartsofthesyllabusforthesecond-andthethird-yearstudents
ofmostoftheuniversities.Assuch,thisvolumetogetherwiththefirstvolumemay
serve as a basic text for the first-, second-, and third-year courses in algebra.
The third volume of the book contains 10 chapters, and it can act as a text for
graduateand advance graduate studentsspecializinginmathematics.This includes
commutative algebra, basics in algebraic geometry, semi-simple Lie algebras,
advancerepresentationtheory,andChevalleygroups.Thetableofcontentsgivesan
idea of the subject matter covered in the book.
There is no prerequisite essential for the book except, occasionally, in some
illustrationsandexercises,someamountofcalculus,geometry,ortopologymaybe
needed. An attempt to follow the logical ordering has been made throughout
the book.
My teacher (Late) Prof. B.L. Sharma, my colleague at the University of
Allahabad, my friend Dr. H.S. Tripathi, my students Prof. R.P. Shukla, Prof.
Shivdatt,Dr.BrajeshKumarSharma,Mr.SwapnilSrivastava,Dr.AkhileshYadav,
Dr. Vivek Jain, Dr. Vipul Kakkar, and above all, the mathematics students of the
University of Allahabad had always been the motivating force for me to write a
book. Without their continuous insistence, it would have not come in the present
form. I wish to express my warmest thanks to all of them.
Harish-Chandra Research Institute (HRI), Allahabad, has always been a great
sourceformetolearnmoreandmoremathematics.Iwishtoexpressmydeepsense
of appreciation and thanks to HRI for providing me all infrastructural facilities to
write these volumes.
Lastbutnotleast,IwishtoexpressmythankstomywifeVeenaSrivastavawho
had always been helpful in this endeavor.
Inspiteofallcare,somemistakesandmisprintsmighthavecreptinandescaped
myattention.Ishallbegratefultoanysuchattention.Criticismsandsuggestionsfor
the improvement of the book will be appreciated and gratefully acknowledged.
Allahabad, India Ramji Lal
April 2017
Contents
1 Vector Spaces ... .... .... ..... .... .... .... .... .... ..... .. 1
1.1 Concept of a Field... ..... .... .... .... .... .... ..... .. 1
1.2 Concept of a Vector Space (Linear Space).. .... .... ..... .. 7
1.3 Subspaces.. .... .... ..... .... .... .... .... .... ..... .. 11
1.4 Basis and Dimension. ..... .... .... .... .... .... ..... .. 16
1.5 Direct Sum of Vector Spaces, Quotient of a Vector Space .. .. 23
2 Matrices and Linear Equations.. .... .... .... .... .... ..... .. 31
2.1 Matrices and Their Algebra. .... .... .... .... .... ..... .. 31
2.2 Types of Matrices ... ..... .... .... .... .... .... ..... .. 35
2.3 System of Linear Equations. .... .... .... .... .... ..... .. 40
2.4 Gauss Elimination, Elementary Operations, Rank,
and Nullity. .... .... ..... .... .... .... .... .... ..... .. 43
2.5 LU Factorization .... ..... .... .... .... .... .... ..... .. 58
2.6 Equivalence of Matrices, Normal Form.... .... .... ..... .. 60
2.7 Congruent Reduction of Symmetric Matrices.... .... ..... .. 65
3 Linear Transformations... ..... .... .... .... .... .... ..... .. 73
3.1 Definition and Examples ... .... .... .... .... .... ..... .. 73
3.2 Isomorphism Theorems .... .... .... .... .... .... ..... .. 75
3.3 Space of Linear Transformations, Dual Spaces .. .... ..... .. 79
3.4 Rank and Nullity.... ..... .... .... .... .... .... ..... .. 83
3.5 Matrix Representations of Linear Transformations.... ..... .. 85
3.6 Effect of Change of Bases on Matrix Representation.. ..... .. 88
4 Inner Product Spaces. .... ..... .... .... .... .... .... ..... .. 97
4.1 Definition, Examples, and Basic Properties . .... .... ..... .. 97
4.2 Gram–Schmidt Process .... .... .... .... .... .... ..... .. 107
4.3 Orthogonal Projection, Shortest Distance... .... .... ..... .. 112
4.4 Isometries and Rigid Motions ... .... .... .... .... ..... .. 120
ix
x Contents
5 Determinants and Forms .. ..... .... .... .... .... .... ..... .. 131
5.1 Determinant of a Matrix.... .... .... .... .... .... ..... .. 131
5.2 Permutations ... .... ..... .... .... .... .... .... ..... .. 135
5.3 Alternating Forms, Determinant of an Endomorphism. ..... .. 139
5.4 Invariant Subspaces, Eigenvalues. .... .... .... .... ..... .. 150
5.5 Spectral Theorem, and Orthogonal Reduction ... .... ..... .. 159
5.6 Bilinear and Quadratic Forms ... .... .... .... .... ..... .. 176
6 Canonical Forms, Jordan and Rational Forms.. .... .... ..... .. 195
6.1 Concept of a Module over a Ring .... .... .... .... ..... .. 195
6.2 Modules over P.I.D .. ..... .... .... .... .... .... ..... .. 203
6.3 Rational and Jordan Forms . .... .... .... .... .... ..... .. 214
7 General Linear Algebra... ..... .... .... .... .... .... ..... .. 229
7.1 Noetherian Rings and Modules .. .... .... .... .... ..... .. 229
7.2 Free, Projective, and Injective Modules.... .... .... ..... .. 234
7.3 Tensor Product and Exterior Power... .... .... .... ..... .. 250
7.4 Lower K-theory. .... ..... .... .... .... .... .... ..... .. 258
8 Field Theory, Galois Theory .... .... .... .... .... .... ..... .. 265
8.1 Field Extensions. .... ..... .... .... .... .... .... ..... .. 265
8.2 Galois Extensions.... ..... .... .... .... .... .... ..... .. 275
8.3 Splitting Field, Normal Extensions.... .... .... .... ..... .. 284
8.4 Separable Extensions. ..... .... .... .... .... .... ..... .. 294
8.5 Fundamental Theorem of Galois Theory ... .... .... ..... .. 305
8.6 Cyclotomic Extensions..... .... .... .... .... .... ..... .. 311
8.7 Geometric Constructions ... .... .... .... .... .... ..... .. 318
8.8 Galois Theory of Equation.. .... .... .... .... .... ..... .. 324
9 Representation Theory of Finite Groups... .... .... .... ..... .. 331
9.1 Semi-simple Rings and Modules . .... .... .... .... ..... .. 331
9.2 Representations and Group Algebras.. .... .... .... ..... .. 346
9.3 Characters, Orthogonality Relations... .... .... .... ..... .. 351
9.4 Induced Representations.... .... .... .... .... .... ..... .. 361
10 Group Extensions and Schur Multiplier... .... .... .... ..... .. 367
10.1 Schreier Group Extensions.. .... .... .... .... .... ..... .. 368
10.2 Obstructions and Extensions .... .... .... .... .... ..... .. 391
10.3 Central Extensions, Schur Multiplier .. .... .... .... ..... .. 398
10.4 Lower K-Theory Revisited.. .... .... .... .... .... ..... .. 418
Bibliography.... .... .... .... ..... .... .... .... .... .... ..... .. 427
Index.. .... .... .... .... .... ..... .... .... .... .... .... ..... .. 429