Basic abstract algebra: for graduate students and advanced undergraduates

309 Pages·2006·2.34 MB·English

by  Robert B. Ash

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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.
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.
Thesetworequirementsareinopposition, anditismyjobasexpositortotrytoresolvethe
conflict.
My primary goal is to help the reader learn the subject, and there are times when
informal or intuitive reasoning leads to greater understanding than a formal proof. In the
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
many pressing matters to cope with, may not be the appropriate place, 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.
Solutionstoallproblemsareincludedinthetext; inmyexperience, moststudentsfind
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 typical
studentwillhaveseensomebutnotallofthismaterialinanundergraduatealgebracourse.
[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-
ablewithelementaryrowandcolumnoperations, andthiswillsignificantlyaidthelearning
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¨ldertheoremandtosolvable
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 pre-
sentation. 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
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directcalculationofaGaloisgroup,weproceedtofinitefields,whichareofgreatimportance
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.
Intheremainingchapters,webegintoapplytheresultsandmethodsofabstractalgebra
torelatedareas. Thetitleofeachchapterbeginswith“Introducing...”,andtheareastobe
introduced are algebraic number theory, algebraic geometry, noncommutative algebra and
homological algebra (including categories and functors).
Algebraic number theory and algebraic geometry are the two major areas that use the
tools of commutative algebra (the theory of commutative rings). In Chapter 7, after an
exampleshowinghowalgebracanbeappliedinnumbertheory, weassemblesomealgebraic
equipment: integralextensions,norms,traces,discriminants,NoetherianandArtinianmod-
ulesandrings. Wethenprovethefundamentaltheoremonuniquefactorizationofidealsina
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
equivalentversionsofthenullstellensatzaregiven,aswellassomecorollarieswithgeometric
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-
son’stheorem,combinetoyieldWedderburn’stheoremonthestructureofsemisimplerings.
We indicate the precise connection between the two popular definitions of simple ring in
the literature. After an informal introduction to group representations, Maschke’s theorem
on semisimplicity of modules over the group algebra is proved. The introduction of the
Jacobson radical gives more insight into the structure of rings and modules. The chapter
endswiththeHopkins-LevitzkitheoremthatanArtinianringisNoetherian, andtheuseful
lemma of Nakayama.
In Chapter 10, we introduce some of the tools of homological algebra. Waiting until
thelastchapterforthisisadeliberatedecision. Studentsneedasmuchexposureaspossible
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
algebrasequence. Wedothistohelpstudentscopewiththemassiveformalmachinerythat
makesitsodifficulttogainaworkingknowledgeofthisarea. Weconcentrateontheresults
that are most useful in applications: the long exact homology sequence and the properties
ofthederivedfunctorsTorandExt. Thereisacompleteproofofthesnakelemma, ararity
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
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whenoneislearningasubject,formalismoftenpreventsunderstanding. Themostimportant
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, Ihaveattemptedtowriteatextofmanageablesize, whichcanbereadbysudents,
including those working independently.
Another goal is to help the student reach an advanced level as quickly and efficiently
as 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
formalproof. Inallcases, Igiveexplicitreferenceswherethedetailscanbefound. Onecan
arguethatlearningtowriteformalproofsisanessentialpartofthestudent’smathematical
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 (cid:1)c 2000, by Robert B. Ash
Paper or electronic copies for noncommercial use may be made freely without explicit per-
mission of the author. All other rights are reserved.
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ABSTRACT ALGEBA: 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
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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
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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
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Chapter 0 PREREQUISITES
AlltopicslistedinthischapterarecoveredinA Primer of Abstract MathematicsbyRobert
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
[We allow negative primes, so that, for example, -17 is prime. This is consistent with the
general definition of prime element in an integral domain; see Section 2.6.]
0.1.4 The Integers Modulo m If a and b are integers and m is a positive integer ≥ 2,
we write a ≡ b mod m, 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 mod m. Residue classes can be added, subtracted and multiplied
consistently by choosing a representative from each class, performing the appropriate oper-
ation, and calculating the residue class of the result. The collection Z of residue classes
m
mod m forms a commutative ring under addition and multiplication. Z is a field if and
m
only if m is prime. (The general definitions of ring, integral domain and field are given in
Section 2.1.)
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) If d=gcd(a,b), the greatest common divisor of a and b, then a/d and b/d are relatively
prime.
(6) If ax≡ay mod m and d=gcd(a,m), then x≡y mod m/d.
(7) If a divides b for i = 1,...,r, and a and a are relatively prime whenever i (cid:6)= j, then
i i j
the product a ···a divides b.
1 r
(8) The product of two integers is their greatest common divisor times their least common
multiple.
0.1.6ChineseRemainderTheorem Ifm ,...,m arerelativelyprimeinpairs,thenthe
1 r
system of simultaneous equations x ≡ b mod m ,j = 1,...,r, has a solution for arbitrary
j j
integersb . Thesetofsolutionsformsasingleresidueclassmodm=m ···m ,sothatthere
j 1 r
is a unique solution mod m.
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
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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
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.
0.2.4MaximumPrinciple IfT isanychain(totallyorderedsubset)ofapartiallyordered
set S, 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
i
element of each S . Formally, there is a function f whose domain is I such that f(i) ∈ S
i i
for all i∈I.
The well-ordering principle, the maximum principle, Zorn’s lemma, and the axiom of
choice are equivalent in the sense that if any one of these statements is added to the basic
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 eachS
i
must be accompanied by an explicit algorithm. The idea is appealing, but its acceptance
resultsinlargeareasofinterestingandusefulmathematicsbeingtossedontothescrapheap.
So at present, the mathematical mainstream embraces the axiom of choice, Zorn’s lemma
et al.
0.2.7 Proof by Transfinite Induction To prove that statement P holds for all i in the
i
well-ordered set I, we do the following:
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 hypothesis),
j
prove P .
i
It follows that P is true for all i.
i
0.2.8 We say that the size of the set A is less than or equal to the size of B (notation
A≤ B) if there is an injective map from A to B. We say that A and B have the same size
s
(A= B) if there is a bijection between A and B.
s
0.2.9 Schr¨oder-Bernstein Theorem If A ≤ B and B ≤ A, then A = B. (This can
s s s
be proved without the axiom of choice.)
0.2.10 Using (0.2.9), one can show that if sets of the same size are called equivalent, then
≤ on equivalence classes is a partial ordering. It follows with the aid of Zorn’s lemma that
s
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the ordering is total. The equivalence class of a set A, written |A|, is called the cardinal
numberorcardinalityofA. Inpractice,weusuallyidentify|A|withanyconvenientmember
of the equivalence class, such as A itself.
0.2.11 ForanysetA, wecanalwaysproduceasetofgreatercardinality, namelythepower
set 2A, that is, the collection of all subsets of A.
0.2.12 Define addition and multiplication of cardinal numbers by |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 .
0 0 0 0 0 0
More 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 If A is an infinite set, then A and the set of all finite subsets of A have the same
cardinality.
0.3 Linear Algebra
Itisnotfeasibletolistallresultspresentedinanundergraduatecourseinlinearalgebra.
Instead, here is a list of topics that are covered in a typical course.
1. Sums, products, transposes, inverses of matrices; symmetric matrices.
2. Elementary row and column operations; reduction to echelon form.
3. Determinants: evaluation by Laplace expansion and Cramer’s rule.
4. Vector spaces over a field; subspaces, linear independence and bases.
5. Rank of a matrix; homogeneous and nonhomogeneous linear equations.
6. Null space and range of a matrix; the dimension theorem.
7. Linear transformations and their representation by matrices.
8. Coordinates and matrices under change of basis.
9. Inner product spaces and the projection theorem.
10. Eigenvalues and eigenvectors; diagonalization of matrices with distinct eigenvalues,
symmetric and Hermitian matrices.
11. Quadratic forms.
A more advanced course might cover the following topics:
12. Generalized eigenvectors and the Jordan canonical form.
13. The minimal and characteristic polynomials of a matrix; Cayley-Hamilton theorem.
14. The adjoint of a linear operator.
15. Projection operators.
16. Normal operators and the spectral theorem.