Abstract Algebra

Springer Undergraduate Mathematics Series

Gregory T. Lee

Abstract

Algebra

An Introductory Course

Springer Undergraduate Mathematics Series

Advisory Board

M.A.J. Chaplain, University of St. Andrews

A. MacIntyre, Queen Mary University of London

S. Scott, King’s College London

N. Snashall, University of Leicester

E. Süli, University of Oxford

M.R. Tehranchi, University of Cambridge

J.F. Toland, University of Bath

More information about this series at http://www.springer.com/series/3423

Gregory T. Lee

Abstract Algebra

An Introductory Course

123

Gregory T. Lee

Department of Mathematical Sciences

Lakehead University

Thunder Bay, ON

Canada

ISSN 1615-2085 ISSN 2197-4144 (electronic)

Springer Undergraduate Mathematics Series

ISBN 978-3-319-77648-4 ISBN 978-3-319-77649-1 (eBook)

https://doi.org/10.1007/978-3-319-77649-1

Library of Congress Control Number: 2018935845

Mathematics Subject Classification (2010): 20-01, 16-01, 12-01

© Springer International Publishing AG, part of Springer Nature 2018

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In memory of my father

Preface

This book is intended for students encountering the beautiful subject of abstract

algebra for the first time. My goal here is to provide a text that is suitable for you,

whether you plan to take only a single course in abstract algebra, or to carry on to

more advanced courses at the senior undergraduate and graduate levels. Naturally, I

wish to encourage you to study the subject further and to ensure that you are

prepared if you do so.

At many universities, including my own, abstract algebra is the first serious

proof-based course taken by mathematics majors. While it is quite possible to get

through, let us say, a course in calculus simply by memorizing a list of rules and

applying them correctly, without really understanding why anything works, such an

approach would be disastrous here. To be sure, you must carefully learn the defi￾nitions and the statements of theorems, but that is nowhere near sufficient. In order

to master the material, you need to understand the proofs and then be able to prove

things yourself. This book contains hundreds of problems, and I cannot stress

strongly enough the need to solve as many of them as you can. Do not be dis￾couraged if you cannot get all of them! Some are very difficult. But try to figure out

as many as you can. You will only learn by getting your hands dirty.

As different universities have different sequences of courses, I am not assuming

any prerequisites beyond the high school level. Most of the material in Part I would

be covered in a typical course on discrete mathematics. Even if you have had such a

course, I urge you to read through it. In particular, you absolutely must understand

equivalence relations and equivalence classes thoroughly. (In my experience, many

students have trouble with these concepts.) From time to time, throughout Parts II

and III, some examples involving matrices or complex numbers appear. These can

be bypassed if you have not studied linear algebra or complex numbers, but in any

case, the material you need to know is not difficult and is discussed in the

appendices. In Part IV, it is necessary to know some linear algebra, but all of the

theorems used are proved in the text.

vii

The fundamental results about groups are covered in Chaps. 3 and 4, those about

rings are in Chaps. 8 and 9, and the introductory theorems concerning fields and

polynomials are found in Chap. 11. I think that these chapters are essential in any

course. Beyond that, there is a fair amount of flexibility in the choice of topics.

I confess my first encounter with abstract algebra was a joyous experience.

I found (and still find!) the subject fascinating, and I will consider the time I put into

this book well spent if you emerge with an appreciation for the field.

I would like to thank Lynn Brandon and Anne-Kathrin Birchley-Brun at

Springer for their help in making this book a reality. Also, thanks to the reviewers

for their many useful suggestions. I thank my wife and family for their ongoing

support. Finally, thanks to my teacher, Prof. Sudarshan Sehgal, both for his advice

concerning this book and for all of his help over the years.

Thunder Bay, ON, Canada Gregory T. Lee

viii Preface

Contents

Part I Preliminaries

1 Relations and Functions ................................. 3

1.1 Sets and Set Operations.............................. 3

1.2 Relations ........................................ 5

1.3 Equivalence Relations ............................... 6

1.4 Functions ........................................ 10

2 The Integers and Modular Arithmetic ...................... 15

2.1 Induction and Well Ordering .......................... 15

2.2 Divisibility ....................................... 19

2.3 Prime Factorization ................................. 24

2.4 Properties of the Integers............................. 26

2.5 Modular Arithmetic ................................. 27

Part II Groups

3 Introduction to Groups .................................. 35

3.1 An Important Example .............................. 35

3.2 Groups .......................................... 38

3.3 A Few Basic Properties.............................. 42

3.4 Powers and Orders ................................. 44

3.5 Subgroups ....................................... 48

3.6 Cyclic Groups .................................... 54

3.7 Cosets and Lagrange’s Theorem ....................... 57

4 Factor Groups and Homomorphisms ....................... 61

4.1 Normal Subgroups ................................. 61

4.2 Factor Groups..................................... 65

4.3 Homomorphisms................................... 69

4.4 Isomorphisms ..................................... 72

ix

4.5 The Isomorphism Theorems for Groups .................. 78

4.6 Automorphisms ................................... 81

5 Direct Products and the Classification of Finite

Abelian Groups........................................ 85

5.1 Direct Products .................................... 85

5.2 The Fundamental Theorem of Finite Abelian Groups ........ 88

5.3 Elementary Divisors and Invariant Factors ................ 93

5.4 A Word About Infinite Abelian Groups .................. 97

6 Symmetric and Alternating Groups ........................ 101

6.1 The Symmetric Group and Cycle Notation ................ 101

6.2 Transpositions and the Alternating Group ................. 105

6.3 The Simplicity of the Alternating Group ................. 108

7 The Sylow Theorems.................................... 115

7.1 Normalizers and Centralizers .......................... 115

7.2 Conjugacy and the Class Equation ...................... 119

7.3 The Three Sylow Theorems........................... 122

7.4 Applying the Sylow Theorems......................... 125

7.5 Classification of the Groups of Small Order ............... 128

Part III Rings

8 Introduction to Rings ................................... 135

8.1 Rings ........................................... 135

8.2 Basic Properties of Rings ............................ 138

8.3 Subrings......................................... 140

8.4 Integral Domains and Fields .......................... 142

8.5 The Characteristic of a Ring .......................... 146

9 Ideals, Factor Rings and Homomorphisms ................... 149

9.1 Ideals ........................................... 149

9.2 Factor Rings...................................... 152

9.3 Ring Homomorphisms .............................. 155

9.4 Isomorphisms and Automorphisms...................... 159

9.5 Isomorphism Theorems for Rings ...................... 165

9.6 Prime and Maximal Ideals............................ 167

10 Special Types of Domains ................................ 171

10.1 Polynomial Rings .................................. 171

10.2 Euclidean Domains ................................. 176

10.3 Principal Ideal Domains ............................. 182

10.4 Unique Factorization Domains......................... 185

Reference ............................................. 188

x Contents

Part IV Fields and Polynomials

11 Irreducible Polynomials ................................. 191

11.1 Irreducibility and Roots.............................. 191

11.2 Irreducibility over the Rationals ........................ 195

11.3 Irreducibility over the Real and Complex Numbers.......... 200

11.4 Irreducibility over Finite Fields ........................ 202

Reference ............................................. 205

12 Vector Spaces and Field Extensions ........................ 207

12.1 Vector Spaces..................................... 207

12.2 Basis and Dimension ............................... 210

12.3 Field Extensions ................................... 215

12.4 Splitting Fields .................................... 221

12.5 Applications to Finite Fields .......................... 225

Reference ............................................. 229

Part V Applications

13 Public Key Cryptography ................................ 233

13.1 Private Key Cryptography ............................ 233

13.2 The RSA Scheme .................................. 236

14 Straightedge and Compass Constructions .................... 241

14.1 Three Ancient Problems ............................. 241

14.2 The Connection to Field Extensions..................... 244

14.3 Proof of the Impossibility of the Problems ................ 250

Appendix A: The Complex Numbers............................. 253

Appendix B: Matrix Algebra ................................... 257

Solutions ................................................... 263

Index ...................................................... 297

Contents xi

Part I

Preliminaries

Chapter 1

Relations and Functions

We begin by introducing some basic notation and terminology. Then we discuss

relations and, in particular, equivalence relations, which we shall see several times

throughout the book. In the final section, we talk about various sorts of functions.

1.1 Sets and Set Operations

A set is a collection of objects. We will see many sorts of sets throughout this course.

Perhaps the most common will be sets of numbers. For instance, we have the set of

natural numbers,

N = {1, 2, 3,...},

the set of integers,

Z = {..., −2, −1, 0, 1, 2,...}

and the set of rational numbers

Q =

a

b

: a, b ∈ Z, b = 0

.

We also write R for the set of real numbers and C for the set of complex numbers.

But sets do not necessarily consist of numbers. Indeed, we can consider the set of

all letters of the alphabet, the set of all polynomials with even integers as coefficients

or the set of all lines in the plane with positive slope.

The objects in a set are called its elements. We write a ∈ S if a is an element of a

set S. Thus, −3 ∈ Z but −3 ∈/ N. The set with no elements is called the empty set,

and denoted ∅. Any other set is said to be nonempty.

© Springer International Publishing AG, part of Springer Nature 2018

G. T. Lee, Abstract Algebra, Springer Undergraduate Mathematics Series,

https://doi.org/10.1007/978-3-319-77649-1_1

3

4 1 Relations and Functions

If S and T are sets, then we say that S is a subset of T , and write S ⊆ T , if every

element of S is also an element of T . Of course, S ⊆ S. We say that S is a proper

subset of T , and write S T , if S ⊆ T but S = T . Thus, it is certainly true that

N ⊆ Z, but we can be more precise and write NZ.

For any two sets S and T , their intersection, S ∩ T , is the set of all elements that

lie in S and T simultaneously.

Example 1.1. Let S = {1, 2, 3, 4, 5} and T = {2, 4, 6, 8, 10}. Then S ∩ T = {2, 4}.

We can extend this notion to the intersection of an arbitrary collection of sets. If

I is a nonempty set and, for each i ∈ I, we have a set Ti , then we write 

i∈I Ti for

the set of elements that lie in all of the Ti simultaneously.

Example 1.2. For each q ∈ Q, let Tq = {r ∈ R : r < 2q }. Then 

q∈Q Tq = {r ∈ R :

r ≤ 0}.

Also, for any sets S and T , their union, S ∪ T , is the set of all elements that lie

in S or T (or both).

Example 1.3. Using the same S and T as in Example 1.1, we have

S ∪ T = {1, 2, 3, 4, 5, 6, 8, 10}.

Furthermore, if I is a nonempty set and we have a set Ti for each i ∈ I, then we

write 

i∈I Ti for the union of all of the Ti ; that is, the set of all elements that lie in

at least one of the Ti .

Example 1.4. If we use the same sets Tq as in Example 1.2, we have 

q∈Q Tq = R.

In addition, for any two sets S and T , the set difference (or relative complement)

is the set S\T = {a ∈ S : a ∈/ T }.

Example 1.5. Once again using S and T as in Example 1.1, we have S\T = {1, 3, 5}.

We will need one more definition. The following construction is named after René

Descartes.

Definition 1.1. Let S and T be any sets. Then the Cartesian product S × T is the

set of all ordered pairs (s, t), with s ∈ S and t ∈ T .

Example 1.6. Let S = {1, 2, 3} and T = {2, 3}. Then

S × T = {(1, 2), (1, 3), (2, 2), (2, 3), (3, 2), (3, 3)}.

There is also a Cartesian product of finitely many sets. For any sets T1, T2,..., Tn,

we let T1 × T2 ×···× Tn be the set of all ordered n-tuples(t1, t2,...,tn), with ti ∈ Ti

for all i.

1.1 Sets and Set Operations 5

Example 1.7. Let T1 = {1, 2}, T2 = {a, b} and T3 = {2, 3}. Then T1 × T2 × T3 is

the set

{(1, a, 2), (1, a, 3), (1, b, 2), (1, b, 3), (2, a, 2), (2, a, 3), (2, b, 2), (2, b, 3)}.

Exercises

1.1. Let S = {1, 2, 3} and T = {3, 4}. Find S ∩ T , S ∪ T , S\T , T \S and S × T .

1.2. Let R = {a, b, c}, S = {a, c, d} and T = {c, e, f }. Find R ∩ S, R ∩ (S\T ),

S ∪ T , S ∩ (R ∪ T ) and R × S.

1.3. Let R, S and T be sets with R ⊆ S. Show that R ∪ T ⊆ S ∪ T .

1.4. Let S = {1, 2,..., n}, for some positive integer n. Show that S has 2n subsets.

1.5. Let R, S and T be any sets. Show that R ∪ (S ∩ T ) = (R ∪ S) ∩ (R ∪ T ).

1.6. For each positive integer n, let Tn = { a

n : a ∈ Z}.

1. What is ∞

n=1 Tn?

2. What is ∞

n=1 Tn?

1.2 Relations

We are going to use relations (in particular, the equivalence relations and functions

that we will see in the next two sections) quite a few times in this course.

Definition 1.2. Let S and T be sets. Then a relation from S to T is a subset ρ of

S × T . If s ∈ S and t ∈ T , then we write sρt if (s, t) ∈ ρ; otherwise, we write s ρ t.

In particular, a relation on S is a relation from S to S.

Example 1.8. Let S = {1, 2, 3} and T = {1, 2, 3, 4}. Define a relation ρ from S to T

via sρt if and only if st 2 ≤ 4. Then ρ = {(1, 1), (1, 2), (2, 1), (3, 1)}. In particular,

3ρ1 but 1 ρ 3.

We will focus on relations on a set. Let us discuss a few properties enjoyed by

some relations.

Definition 1.3. Let ρ be a relation on S. We say that ρ is reflexive if aρa for all

a ∈ S.

Example 1.9. On Z, the relation ≤ is reflexive, but < is not. Indeed, a ≤ a for all

integers a, but 1 is not less than 1.

Definition 1.4. A relation ρ on a set S is symmetric if aρb implies bρa.