Abstract Algebra Robert B. Ash 295p

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Abstract Algebra: The Basic Graduate Year

Robert B. Ash

PREFACE

This is a text for the basic graduate sequence in abstract algebra, offered by most

universities. We study fundamental algebraic structures, namely groups, rings, fields and

modules, and maps between these structures. The techniques are used in many areas of

mathematics, and there are applications to physics, engineering and computer science as

well. In addition, I have attempted to communicate the intrinsic beauty of the subject.

text, there are three types of informal arguments:

1. The concrete or numerical example with all features of the general case. Here, the

example indicates how the proof should go, and the formalization amounts to substituting

Greek letters for numbers. There is no essential loss of rigor in the informal version.

2. Brief informal surveys of large areas. There are two of these, p-adic numbers and group

representation theory. References are given to books accessible to the beginning graduate

student.

3. Intuitive arguments that replace lengthy formal proofs which do not reveal why a result

is true. In this case, explicit references to a precise formalization are given. I am not saying

that the formal proof should be avoided, just that the basic graduate year, where there are

main text, and this allows for wider coverage as well as reinforcement of the basic ideas.

Chapters 1-4 cover basic properties of groups, rings, fields and modules. The typical

student will have seen some but not all of this material in an undergraduate algebra course.

[It should be possible to base an undergraduate course on Chapters 1-4, traversed at a

suitable pace with detailed coverage of the exercises.] In Chapter 4, the fundamental struc-

ture theorems for finitely generated modules over a principal ideal domain are developed

concretely with the aid of the Smith normal form. Students will undoubtedly be comfort-

able with elementary row and column operations, and this will significantly aid the learning

process.

In Chapter 5, the theme of groups acting on sets leads to a nice application to com-

Chapter 6 goes quickly to the fundamental theorem of Galois theory; this is possible

because the necessary background has been covered in Chapter 3. After some examples of

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direct calculation of a Galois group, we proceed to finite fields, which are of great importance

in applications, and cyclotomic fields, which are fundamental in algebraic number theory.

The Galois group of a cubic is treated in detail, and the quartic is covered in an appendix.

Sections on cyclic and Kummer extensions are followed by Galois’ fundamental theorem on

solvability by radicals. The last section of the chapter deals with transcendental extensions

and transcendence bases.

In the remaining chapters, we begin to apply the results and methods of abstract algebra

to related areas. The title of each chapter begins with “Introducing...”, and the areas to be

introduced are algebraic number theory, algebraic geometry, noncommutative algebra and

Dedekind domain. The chapter concludes with an informal introduction to p-adic numbers

and some ideas from valuation theory.

Chapter 8 begins geometrically with varieties in affine space. This provides motivation

for Hilbert’s fundamental theorems, the basis theorem and the nullstellensatz. Several

equivalent versions of the nullstellensatz are given, as well as some corollaries with geometric

significance. Further geometric considerations lead to the useful algebraic techniques of

localization and primary decomposition. The remainder of the chapter is concerned with

the tensor product and its basic properties.

Chapter 9 begins the study of noncommutative rings and their modules. The basic

theory of simple and semisimple rings and modules, along with Schur’s lemma and Jacob-

lemma of Nakayama.

In Chapter 10, we introduce some of the tools of homological algebra. Waiting until

the last chapter for this is a deliberate decision. Students need as much exposure as possible

to specific algebraic systems before they can appreciate the broad viewpoint of category

theory. Even experienced students may have difficulty absorbing the abstract definitions of

kernel, cokernel, product, coproduct, direct and inverse limit. To aid the reader, functors

are introduced via the familiar examples of hom and tensor. No attempt is made to work

with general abelian categories. Instead, we stay within the category of modules and study

projective, injective and flat modules.

In a supplement, we go much farther into homological algebra than is usual in the basic

but to make most efficient use of the finite amount of time available.

Robert B. Ash

October 2000

Further Remarks

Many mathematicians believe that formalism aids understanding, but I believe that

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when one is learning a subject, formalism often prevents understanding. The most important

skill is the ability to think intuitively. This is true even in a highly abstract field such as

homological algebra. My writing style reflects this view.

Classroom lectures are inherently inefficient. If the pace is slow enough to allow com-

prehension as the lecture is delivered, then very little can be covered. If the pace is fast

enough to allow decent coverage, there will unavoidably be large gaps. Thus the student

must depend on the textbook, and the current trend in algebra is to produce massive en-

cyclopedias, which are likely to be quite discouraging to the beginning graduate student.

Instead, I have attempted to write a text of manageable size, which can be read by sudents,

training. I agree, but the ability to think intuitively is fundamental and must come first.

I would add that the way things are today, there is absolutely no danger that the student

will be insufficiently exposed to formalism and abstraction. In fact there is quite a bit of it

in this book, although not 100 percent.

I offer this text in the hope that it will make the student’s trip through algebra more

enjoyable. I have done my best to avoid gaps in the reasoning. I never use the phrase

“it is easy to see” under any circumstances. I welcome comments and suggestions for

improvement.

Copyright

c

 2000, by Robert B. Ash

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ABSTRACT ALGEBA: THE BASIC GRADUATE YEAR

0.3 Linear Algebra

CHAPTER 1 GROUP FUNDAMENTALS

1.1 Groups and Subgroups

1.2 Permutation Groups

1.3 Cosets, Normal Subgroups and Homomorphisms

1.4 The Isomorphism Theorems

1.5 Direct Products

CHAPTER 2 RING FUNDAMENTALS

2.1 Basic Definitions and Properties

2.2 Ideals, Homomorphisms and Quotient Rings

2.9 Irreducible Polynomials

CHAPTER 3 FIELD FUNDAMENTALS

3.1 Field Extensions

3.2 Splitting Fields

3.3 Algebraic Closures

3.4 Separability

3.5 Normal Extensions

CHAPTER 4 MODULE FUNDAMENTALS

4.1 Modules and Algebras

4.2 The Isomorphism Theorems For Modules

5.6 Composition Series

5.7 Solvable and Nilpotent Groups

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5.8 Generators and Relations

10.5 Projective Modules

10.6 Injective Modules