Abstract algebra, 1st graduate year course

407 Pages·2000·2.839 MB·English

by  Ash R.

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.
Ideally, the reasoning underlying each step of a proof should be completely clear, but the
overall argument should be as brief as possible, allowing a sharp overview of the result.
These two requirements are in opposition, and it is my job as expositor to try to resolve
the conflict.
My primary goal is to help the reader learn the subject, and there are times when
informalorintuitivereasoningleadstogreaterunderstandingthanaformalproof. Inthe
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 substi-
tuting 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
grouprepresentationtheory. Referencesaregiventobooksaccessibletothebeginning
graduate student.
3. Intuitive arguments that replace lengthy formal proofs which do not reveal why a
resultistrue. Inthiscase,explicitreferencestoapreciseformalizationaregiven. Iam
not saying that the formal proof should be avoided, just that the basic graduate year,
wheretherearemanypressingmatterstocopewith,maynotbetheappropriateplace,
especially when the result rather than the proof technique is used in applications.
I would estimate that about 90 percent of the text is written in conventional style,
and I hope that the book will be used as a classroom text as well as a supplementary
reference.
Solutions to all problems are included in the text; in my experience, most students
find this to be a valuable feature. The writing style for the solutions is similar to that
of the 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 typi-
cal student will have seen some but not all of this material in an undergraduate algebra
course. [ItshouldbepossibletobaseanundergraduatecourseonChapters1–4,traversed
at a suitable pace with detailed coverage of the exercises.] In Chapter 4, the fundamental
structure theorems for finitely generated modules over a principal ideal domain are de-
veloped concretely with the aid of the Smith normal form. Students will undoubtedly be
comfortable 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-
binatorics as well as the fundamental Sylow theorems and some results on simple groups.
AnalysisofnormalandsubnormalseriesleadstotheJordan-Ho¨ldertheoremandtosolv-
able and nilpotent groups. The final section, on defining a group by generators and
relations, concentrates on practical cases where the structure of a group can be deduced
from its presentation. Simplicity of the alternating groups and semidirect products are
covered in the exercises.
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 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
coveredinanappendix. SectionsoncyclicandKummerextensionsarefollowedbyGalois’
fundamentaltheoremonsolvabilitybyradicals. Thelastsectionofthechapterdealswith
transcendental extensions and transcendence bases.
In the remaining chapters, we begin to apply the results and methods of abstract
algebratorelatedareas. Thetitleofeachchapterbeginswith“Introducing... ”,andthe
areas to be introduced are algebraic number theory, algebraic geometry, noncommutative
algebra and homological algebra (including categories and functors).
Algebraicnumbertheoryandalgebraicgeometryarethetwomajorareasthatusethe
tools of commutative algebra (the theory of commutative rings). In Chapter 7, after an
exampleshowinghowalgebracanbeappliedinnumbertheory,weassemblesomealgebraic
equipment: integral extensions, norms, traces, discriminants, Noetherian and Artinian
modules and rings. We then prove the fundamental theorem on unique factorization of
ideals in a 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 moti-
vation 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 Ja-
cobson’stheorem,combinetoyieldWedderburn’stheoremonthestructureofsemisimple
rings. We indicate the precise connection between the two popular definitions of simple
ringintheliterature. Afteraninformalintroductiontogrouprepresentations, Maschke’s
theoremonsemisimplicityofmodulesoverthegroupalgebraisproved. Theintroduction
of the Jacobson radical gives more insight into the structure of rings and modules. The
chapterendswiththeHopkins-LevitzkitheoremthatanArtinianringisNoetherian, and
the useful 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.
Inasupplement,wegomuchfartherintohomologicalalgebrathanisusualinthebasic
algebra sequence. We do this to help students cope with the massive formal machinery
that makes it so difficult to gain a working knowledge of this area. We concentrate on
the results that are most useful in applications: the long exact homology sequence and
the properties of the derived functors Tor and Ext. There is a complete proof of the
snake lemma, a rarity in the literature. In this case, going through a long formal proof is
entirely appropriate, because doing so will help improve algebraic skills. The point is not
to avoid difficulties, 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
when one is learning a subject, formalism often prevents understanding. The most im-
portant 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
comprehension 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
massiveencyclopedias,whicharelikelytobequitediscouragingtothebeginninggraduate
student. Instead, I have attempted to write a text of manageable size, which can be read
by students, including those working independently.
Anothergoalistohelpthestudentreachanadvancedlevelasquicklyandefficientlyas
possible. When I omit a lengthy formal argument, it is because I judge that the increase
in algebraic skills is insufficient to justify the time and effort involved in going through
the formal proof. In all cases, I give explicit references where the details can be found.
One can argue that learning to write formal proofs is an essential part of the student’s
mathematical training. I agree, but the ability to think intuitively is fundamental and
mustcomefirst. Iwouldaddthatthewaythingsaretoday, thereisabsolutelynodanger
thatthestudentwillbeinsufficientlyexposedtoformalismandabstraction. Infactthere
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 (cid:1)c 2000, by Robert B. Ash
Paper or electronic copies for noncommercial use may be made freely without explicit
permission of the author. All other rights are reserved.
ABSTRACT ALGEBRA: THE BASIC GRADUATE YEAR
TABLE OF CONTENTS
CHAPTER 0 PREREQUISITES
0.1 Elementary Number Theory
0.2 Set Theory
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
2.3 The Isomorphism Theorems For Rings
2.4 Maximal and Prime Ideals
2.5 Polynomial Rings
2.6 Unique Factorization
2.7 Principal Ideal Domains and Euclidean Domains
2.8 Rings of Fractions
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
4.3 Direct Sums and Free Modules
4.4 Homomorphisms and Matrices
4.5 Smith Normal Form
4.6 Fundamental Structure Theorems
4.7 Exact Sequences and Diagram Chasing
CHAPTER 5 SOME BASIC TECHNIQUES OF GROUP THEORY
5.1 Groups Acting on Sets
5.2 The Orbit-Stabilizer Theorem
5.3 Applications to Combinatorics
5.4 The Sylow Theorems
5.5 Applications of the Sylow Theorems
5.6 Composition Series
5.7 Solvable and Nilpotent Groups
5.8 Generators and Relations
CHAPTER 6 GALOIS THEORY
6.1 Fixed Fields and Galois Groups
6.2 The Fundamental Theorem
6.3 Computing a Galois Group Directly
6.4 Finite Fields
6.5 Cyclotomic Fields
6.6 The Galois Group of a Cubic
6.7 Cyclic and Kummer Extensions
6.8 Solvability by Radicals
6.9 Transcendental Extensions
Appendix to Chapter 6
CHAPTER 7 INTRODUCING ALGEBRAIC NUMBER THEORY
7.1 Integral Extensions
7.2 Quadratic Extensions of the Rationals
7.3 Norms and Traces
7.4 The Discriminant
7.5 Noetherian and Artinian Modules and Rings
7.6 Fractional Ideals
7.7 Unique Factorization of Ideals in a Dedekind Domain
7.8 Some Arithmetic in Dedekind Domains
7.9 p-adic Numbers
CHAPTER 8 INTRODUCING ALGEBRAIC GEOMETRY
8.1 Varieties
8.2 The Hilbert Basis Theorem
8.3 The Nullstellensatz: Preliminaries
8.4 The Nullstellensatz: Equivalent Versions and Proof
8.5 Localization
8.6 Primary Decomposition
8.7 Tensor Product of Modules Over a Commutative Ring
8.8 General Tensor Products
CHAPTER 9 INTRODUCING NONCOMMUTATIVE ALGEBRA
9.1 Semisimple Modules
9.2 Two Key Theorems
9.3 Simple and Semisimple Rings
9.4 Further Properties of Simple Rings, Matrix Rings, and Endomorphisms
9.5 The Structure of Semisimple Rings
9.6 Maschke’s Theorem
9.7 The Jacobson Radical
9.8 Theorems of Hopkins-Levitzki and Nakayama
CHAPTER 10 INTRODUCING HOMOLOGICAL ALGEBRA
10.1 Categories
10.2 Products and Coproducts
10.3 Functors
10.4 Exact Functors
10.5 Projective Modules
10.6 Injective Modules
10.7 Embedding into an Injective Module
10.8 Flat Modules
10.9 Direct and Inverse Limits
Appendix to Chapter 10
SUPPLEMENT
S1 Chain Complexes
S2 The Snake Lemma
S3 The Long Exact Homology Sequence
S4 Projective and Injective Resolutions
S5 Derived Functors
S6 Some Properties of Ext and Tor
S7 Base Change in the Tensor Product
SOLUTIONS TO PROBLEMS
Chapter 0
Prerequisites
All topics listed in this chapter are covered in A Primer of Abstract Mathematics by
Robert B. Ash, MAA 1998.
0.1 Elementary Number Theory
The greatest common divisor of two integers can be found by the Euclidean algorithm,
which is reviewed in the exercises in Section 2.5. Among the important consequences of
the algorithm are the following three results.
0.1.1
If d is the greatest common divisor of a and b, then there are integers s and t such that
sa+tb=d. In particular, if a and b are relatively prime, there are integers s and t such
that sa+tb=1.
0.1.2
If a prime p divides a product a ···a of integers, then p divides at least one a
1 n i
0.1.3 Unique Factorization Theorem
If a is an integer, not 0 or ±1, then
(1) a can be written as a product p ···p of primes.
1 n
(2) If a = p ···p = q ···q , where the p and q are prime, then n = m and, after
1 n 1 m i j
renumbering, p =±q for all i.
i i
[Weallownegativeprimes,sothat,forexample,−17isprime. Thisisconsistentwiththe
general definition of prime element in an integral domain; see Section 2.6.]
1
2 CHAPTER 0. PREREQUISITES
0.1.4 The Integers Modulo m
If a and b are integers and m is a positive integer ≥ 2, we write a ≡ bmodm, and say
that a is congruent to b modulo m, if a−b is divisible by m. Congruence modulo m
is an equivalence relation, and the resulting equivalence classes are called residue classes
modm. Residueclassescanbeadded,subtractedandmultipliedconsistentlybychoosing
a representative from each class, performing the appropriate operation, and calculating
the residue class of the result. The collection Z of residue classes mod m forms a
m
commutative ring under addition and multiplication. Z is a field if and only if m is
m
prime. (Thegeneraldefinitionsofring,integraldomainandfieldaregiveninSection2.1.)
0.1.5
(1) The integer a is relatively prime to m if and only if a is a unit mod m, that is, a has
a multiplicative inverse mod m.
(2) If c divides ab and a and c are relatively prime, then c divides b.
(3) If a and b are relatively prime to m, then ab is relatively prime to m.
(4) If ax≡ay mod m and a is relatively prime to m, then x≡y mod m.
(5) Ifd=gcd(a,b),thegreatestcommondivisorofaandb,thena/dandb/darerelatively
prime.
(6) If ax≡ay mod m and d=gcd(a,m), then x≡y mod m/d.
(7) Ifa dividesbfori=1,...,r,anda anda arerelativelyprimewheneveri(cid:6)=j,then
i i j
the product a ···a divides b.
1 r
(8) Theproductoftwointegersistheirgreatestcommondivisortimestheirleastcommon
multiple.
0.1.6 Chinese Remainder Theorem
If m ,...,m are relatively prime in pairs, then the system of simultaneous equations
1 r
x≡b mod m ,j =1,...,r, has a solution for arbitrary integers b . The set of solutions
j j j
forms a single residue class mod m=m ···m , so that there is a unique solution mod m.
1 r
This result can be derived from the abstract form of the Chinese remainder theorem;
see Section 2.3.
0.1.7 Euler’s Theorem
The Euler phi function is defined by ϕ(n) = the number of integers in {1,...,n} that
are relatively prime to n. For an explicit formula for ϕ(n), see Section 1.1, Problem 13.
Euler’s theorem states that if n≥2 and a is relatively prime to n, then aϕ(n) ≡1 mod n.
0.1.8 Fermat’s Little Theorem
If a is any integer and p is a prime not dividing a, then ap−1 ≡ 1 mod p. Thus for any
integer a and prime p, whether or not p divides a, we have ap ≡a mod p.
For proofs of (0.1.7) and (0.1.8), see (1.3.4).
0.2. SET THEORY 3
0.2 Set Theory
0.2.1
A partial ordering on a set S is a relation on S that is reflexive (x ≤ x for all x ∈ S),
antisymmetric (x≤y and y ≤x implies x=y), and transitive (x≤y and y ≤z implies
x≤z). If for all x,y ∈S, either x≤y or y ≤x, the ordering is total.
0.2.2
A well-ordering on S is a partial ordering such that every nonempty subset A of S has a
smallest element a. (Thus a≤b for every b∈A).
0.2.3 Well-Ordering Principle
Every set can be well-ordered.
0.2.4 Maximum Principle
If T is any chain (totally ordered subset) of a partially ordered setS, then T is contained
in a maximal chain M. (Maximal means that M is not properly contained in a larger
chain.)
0.2.5 Zorn’s Lemma
If S is a nonempty partially ordered set such that every chain of S has an upper bound
in S, then S has a maximal element.
(The element x is an upper bound of the set A if a ≤ x for every a ∈ A. Note that
x need not belong to A, but in the statement of Zorn’s lemma, we require that if A is a
chain of S, then A has an upper bound that actually belongs to S.)
0.2.6 Axiom of Choice
Given any family of nonempty sets S , i ∈ I, we can choose an element of each S .
i i
Formally, there is a function f whose domain is I such that f(i)∈S for all i∈I.
i
The well-ordering principle, the maximum principle, Zorn’s lemma, and the axiom of
choiceareequivalentinthesensethatifanyoneofthesestatementsisaddedtothebasic
axioms of set theory, all the others can be proved. The statements themselves cannot be
proved from the basic axioms. Constructivist mathematics rejects the axiom of choice
and its equivalents. In this philosophy, an assertion that we can choose an element from
each S must be accompanied by an explicit algorithm. The idea is appealing, but its
i
acceptance results in large areas of interesting and useful mathematics being tossed onto
the scrap heap. So at present, the mathematical mainstream embraces the axiom of
choice, Zorn’s lemma et al.
4 CHAPTER 0. PREREQUISITES
0.2.7 Proof by Transfinite Induction
To prove that statement P holds for all i in the well-ordered set I, we do the following:
i
1. Prove the basis step P , where 0 is the smallest element of I.
0
2. If i > 0 and we assume that P holds for all j < i (the transfinite induction
j
hypothesis), prove P .
i
It follows that P is true for all i.
i
0.2.8
WesaythatthesizeofthesetAislessthanorequaltothesizeofB (notationA≤ B)if
s
thereisaninjectivemapfromAtoB. WesaythatAandB havethesamesize(A= B)
s
if there is a bijection between A and B.
0.2.9 Schro¨der-Bernstein Theorem
If A≤ B and B ≤ A, then A= B. (This can be proved without the axiom of choice.)
s s s
0.2.10
Using (0.2.9), one can show that if sets of the same size are called equivalent, then ≤
s
on equivalence classes is a partial ordering. It follows with the aid of Zorn’s lemma that
the ordering is total. The equivalence class of a set A, written |A|, is called the cardinal
number or cardinality of A. In practice, we usually identify |A| with any convenient
member of the equivalence class, such as A itself.
0.2.11
ForanysetA,wecanalwaysproduceasetofgreatercardinality,namelythepowerset 2A,
that is, the collection of all subsets of A.
0.2.12
Defineadditionandmultiplicationofcardinalnumbersby|A|+|B|=|A∪B|and|A||B|=
|A×B|. In defining addition, we assume that A and B are disjoint. (They can always be
disjointized by replacing a∈A by (a,0) and b∈B by (b,1).)
0.2.13
If ℵ is the cardinal number of a countably infinite set, then ℵ +ℵ =ℵ ℵ =ℵ . More
0 0 0 0 0 0
generally,
(a) If α and β are cardinals, with α≤β and β infinite, then α+β =β.
(b) If α(cid:6)=0 (i.e., α is nonempty), α≤β and β is infinite, then αβ =β.
0.2.14
IfAisaninfiniteset,thenAandthesetofallfinitesubsetsofAhavethesamecardinality.