INTRODUCTION TO ABSTRACT ALGEBRA - Department of

317 Pages·2001·1.56 MB·English

INTRODUCTION
TO ABSTRACT
ALGEBRA
Elbert A. Walker
New Mexico State University
Las Cruces, New Mexico, USA
(cid:13)c1998 byElbert A.Walker, New Mexico State University.
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Contents
1 Sets 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Sets and Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 The Integers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Groups 25
2.1 De(cid:12)nitions and Examples . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Subgroups and Cosets . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.5 The Groups Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.6 Direct Products and Semi-direct Products . . . . . . . . . . . . . 63
2.7 Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . 70
3 Vector Spaces 79
3.1 De(cid:12)nitions and Examples . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Homomorphisms of Vector Spaces. . . . . . . . . . . . . . . . . . 88
3.3 Linear Independence and Bases . . . . . . . . . . . . . . . . . . . 94
3.4 Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4 Rings and Modules 111
4.1 De(cid:12)nitions and Examples . . . . . . . . . . . . . . . . . . . . . . 111
4.2 Homomorphisms and Quotient Rings . . . . . . . . . . . . . . . . 118
4.3 Field of Quotients of an Integral Domain. . . . . . . . . . . . . . 126
4.4 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . 130
4.5 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.6 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.7 Modules overPrincipal Ideal Domains . . . . . . . . . . . . . . . 157
iii
iv CONTENTS
5 Linear Transformations 169
5.1 Linear Transformations and Matrices . . . . . . . . . . . . . . . . 169
5.2 The Rational Canonical Form for Matrices. . . . . . . . . . . . . 177
5.3 EigenvectorsandEigenvalues;theJordanCanonicalForm[Eigenvectors
and Eigenvalues] . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.5 Equivalence of Matrices . . . . . . . . . . . . . . . . . . . . . . . 212
5.6 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6 Fields 237
6.1 Sub(cid:12)elds and Extension Fields . . . . . . . . . . . . . . . . . . . 237
6.2 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
6.3 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
6.4 Solvability by Radicals . . . . . . . . . . . . . . . . . . . . . . . . 261
7 Topics from Group Theory 269
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
7.2 The Jordan-H(cid:127)olderTheorem . . . . . . . . . . . . . . . . . . . . 269
7.3 The Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 272
7.4 Solvable and Nilpotent Groups . . . . . . . . . . . . . . . . . . . 275
7.5 The Wedderburn Theorem for Finite Division Rings . . . . . . . 280
7.6 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . 281
8 Topics in Ring Theory 283
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
8.2 Unique Factorization Domains . . . . . . . . . . . . . . . . . . . 284
8.3 The Hilbert Basis Theorem . . . . . . . . . . . . . . . . . . . . . 289
8.4 The Lasker-NoetherDecomposition Theorem . . . . . . . . . . . 291
8.5 Semisimple Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Appendix: Zorn’s Lemma 305
Bibliography 311
Preface
In teaching a beginning course in abstract algebra, one must suppress the urge
to cover a lot of material and to be as general as possible. The diÆculties in
teaching such a course are pedagogical, not mathematical. The subject matter
isabstract,yetitmustbekeptmeaningfulforstudentsmeetingabstractnessfor
perhaps the (cid:12)rst time. It is better for a student to know what a theorem says
than to be able to quote it, produce a proof of it detail by detail, and not have
the faintest notion of what it’s all about. However, careful attention must be
paid to rigor, and sloppy thinking and incoherent writing cannot be tolerated.
Butrigorshouldbe (cid:13)avoredwith understanding. Understandingthe contentof
a theorem is as important as being able to prove it. I have tried to keep these
things in mind while writing this book.
Thespeci(cid:12)csubjectmatterchosenhereisstandard,andthearrangementof
topics is not particularlybizarre. In an algebracourse, I believe one should get
onwithalgebraassoonaspossible. ThisiswhyIhavekeptChapter1toabare
minimum. Ididn’twanttoincludeitatall,but thematerialthereisabsolutely
essentialforChapter2,andthestudents’knowledgeofitisapttobeabithazy.
Other bits of \set theory" will be expounded upon as the need arises. Zorn’s
Lemma and some of its applications are discussed in the Appendix.
Groups arechosenas the (cid:12)rst algebraicsystem to study for severalreasons.
The notion is needed in subsequent topics. The axioms for a group are simpler
thanthosefortheothersystemstobestudied. Suchbasicnotionsashomomor-
phisms and quotient systems appear in their simplest and purest forms. The
student can readily recognize many old friends as abstract groups in concrete
disguise.
The (cid:12)rst topic slated for a thorough study is that of vector spaces. It is
this material that is most universally useful, and it is important to present
it as soon as is practical. In fact, the arrangement of Chapters 3 through 5
is a compromise between mathematical eÆciency and getting the essentials of
linear algebra done. Chapter 4 is where it is because its results are beautifully
applicable in Chapter 5 and contain theorems one would do anyway. Besides,
there are some nice applications of Chapter 3 in Chapter 4. Chapter 6 is basic
and should not be slighted in favor of 7 or 8. A feature of Chapter 7 is an
algebraicproof of the fundamental theorem of algebra.
Therearemanyexercisesinthisbook. Mathematicsisnotaspectatorsport,
and is best learned by doing. The exercises are provided so that students can
testtheirknowledgeofthematerialin thebodyofthetext,practiceconcocting
proofs on their own, and pick up a few additional facts.
vi CONTENTS
There is no need here to extol the virtues of the abstract approach, and
the importance of algebra to the various areas of mathematics. They are well
known to the professional, and will become fully appreciated by the student
only through experience.
Elbert A. Walker
Las Cruces, New Mexico
March, 1986
Chapter 1
Sets
1.1 Introduction
The purpose of this book is to present some of the basic properties of groups,
rings, (cid:12)elds, and vector spaces. However, some preliminaries are necessary.
There are some facts about sets, mappings, equivalence relations, and the like
thatwillbeusedthroughoutthetext,andwhichareindispensable. Thischapter
presentsthose facts. It isnot meant tobe an introduction to \set theory." The
amount of material is kept to a minimum | just what is needed to get started
on algebra.
Afew fundamentalfactsconcerningthe integerswillbeassumed. Theseare
spelled out in Section 1.6.
1.2 Sets and Subsets
Our approach to sets is the naive or intuitive one. An axiomatic approach to
sets here would lead us too far a(cid:12)eld and would contribute nothing toward the
understanding of the basic properties of a (cid:12)eld, for example.
By the word set, we mean any collection of objects. The objects in the
collection are called the elements of the set. Sets will usually be denoted by
capital letters. If s is an element of a set S, we write s S, and if s is not an
2
element of S, we write s = S. For any object s, either s S or s = S. For the
2 2 2
set Zof all integers, we have 5 Z,but 1=2 = Z. A set is determined by its
(cid:0) 2 2
elements. That is, two sets are the same if they have the same elements. Thus
two sets S and T are equal, written S = T, if s S implies s T, and t T
2 2 2
implies t S.
2
To specify a set, we must tell what its elements are, and this may be done
in various ways. One way is just to write out an ordinary sentence which does
that. For example, we could say \S is the set whose elements are the integers
1, 2, and 3." A shorthand has been developed for this sort of thing, however.
S = 1;2;3 means that S is the set whose elements are 1, 2, and 3. Thus,
f g
1
2 CHAPTER 1. SETS
one way to de(cid:12)ne a set is to list its elements and put braces around the list.
If S is a big set, this procedure can be cumbersome. For example, it would be
tiresome to describe the set of positive integers less that 1,000,000in this way.
The sentence we have just written would be more eÆcient. However, there is
a convention that is universally used that is most convenient. Suppose S is a
set,andthatit istheset ofallelements swhichsatisfysomepropertyP. Then
S = s:s satis(cid:12)esproperty P meansthatS isthesetofallelementsswhich
f g
enjoy property P. For example, S = n:n is an integer, 0<n<1;000;000
f g
is the set of all positive integers less than 1;000;000. Here, property P is the
propertyofbeingapositiveintegerlessthan1;000;000. Ifwealreadyknewthat
Zwas the set of integers, we could write S = n: n Z,0<n <1;000;000 .
f 2 g
This is sometimes written S = n Z: 0 < n < 1;000;000 . The set of even
f 2 g
integers could be written in any of the following ways:
a. n Z:n is even ,
f 2 g
b. n:n is an even integer ,
f g
c. 2n:n Z ,or
f 2 g
d. n:n=2m for some m Z .
f 2 g
The letterZwill beusedthroughoutthis booktodenotetheset ofintegers.
Suppose S and T are sets. If every element of S is an element of T, then
S is a subset of T. This is denoted S T, or T S. Note that S T does
(cid:26) (cid:27) (cid:26)
not rule out the possibility that S = T. The symbol S $ T means S T and
(cid:26)
S =T,butwewillnothavemuchoccasiontousethislatternotation. IfS $T,
6
then S is a proper subset of T. Note that S = T implies both S T and
(cid:26)
T S. Also, if S T and T S, then S =T. Thus, S =T if andonlyif both
(cid:26) (cid:26) (cid:26)
S T and T S.
(cid:26) (cid:26)
It is convenient to allow the possibility that a set have no elements. Such a
set is called the empty set. There is only one such, since, if S and T are both
empty, then S T and T S, so S = T. The empty set is denoted . It has
(cid:26) (cid:26) ;
the property that S for any set S.
;(cid:26)
There are various ways to make new sets from old. For example, given sets
S and T, there are various ways to associate with them a third set. The more
common ones are listed below.
De(cid:12)nition 1.2.1 Let S and T be sets.
a. S T = x:x S or x T . This is the union of S and T.
[ f 2 2 g
b. S T = x:x S and x T . This is the intersection of S and T.
\ f 2 2 g
c. S T = x : x S and x = T . This is the di(cid:11)erence S minus T, or
n f 2 2 g
the complement of T in S.
1.2. SETS AND SUBSETS 3
For example, if R denotes the set of realnumbers, S = x R :0<x<2 ,
f 2 g
and T = x R:1 x 3 , then
f 2 (cid:20) (cid:20) g
S T = x R :0<x 3 ,
[ f 2 (cid:20) g
S T = x R :1 x<2 ,
\ f 2 (cid:20) g
and
S T = x R :0<x<1 :
n f 2 g
Let’s note some properties of , , and . First, keep in mind what they
[ \ n
mean. S T is that set consisting of all the elements of S along with all the
[
elements of T; S T is that set consisting of the elements that are in both S
\
and T;. and S T is that set consisting of the elements that are in S but not in
n
T.
Let A, B, and C be sets. The following are immediate from the de(cid:12)nitions.
a. A A=A; A A=A; A A= .
[ \ n ;
b. A B =B A; A B =B A.
[ [ \ \
c. (A B) C =A (B C); (A B) C =A (B C).
[ [ [ [ \ \ \ \
d. A =A; A = .
[; \; ;
Less obvious are the following, which we will prove.
e. A (B C)=(A B) (A C).
\ [ \ [ \
f. A (B C)=(A B) (A C).
[ \ [ \ [
g. A (B C)=(A B) (A C).
n \ n [ n
To prove(e), we must show that each element of A (B C) is an element
\ [
of (A B) (A C), and that eachelement of (A B) (A C) is an element
\ [ \ \ \ \
of A (B C). Suppose that x A (B C). Then x A and x B C.
\ [ 2 \ [ 2 2 [
Therefore x A B or x A C. Thus x (A B) (A C).
2 \ 2 \ 2 \ [ \
Now suppose that x (A B) (A C). Then x A B or x A C.
2 \ [ \ 2 \ 2 \
Thusx A, andx B C. Hence x A (B C). Therefore(e) is true. One
2 2 [ 2 \ [
can prove (f) in a similar fashion, and you are asked to do so in Problem 7 at
the end of this section.
Wenowprove(g). Supposethatx A (B C). Thenx Aandx = (B C).
2 n \ 2 2 \
Thus, x A B or x A C. Therefore, x (A B) (A C). Suppose that
2 n 2 n 2 n [ n
x (A B) (A C). Then x A B or x A C. Thus, x A and x = B C.
2 n [ n 2 n 2 n 2 2 [
Therefore x A (B C), and this proves (g).
2 n \
When B and C are subsets of A, (g) is one of the De Morgan Rules.
0
Problem 14 (page 14) is another. For a subset S of A, letting S denote A S,
n
these two rules become
0 0 0
(B C) =B C ,
\ [
4 CHAPTER 1. SETS
and
0 0 0
(B C) =B C :
[ \
Let’s back up and consider (e). It asserts that
A (B C)=(A B) (A C),
\ [ \ [ \
with nothing said about the sets A, B, and C. That is, this equality holds for
all sets A, B, and C. That is what we proved. In our proof, nothing was used
about A, B, and C except that they were sets. No matter what sets we take
for A, B, and C, the equality (e) holds. For example, let B, C, and D be any
sets, and let A = B D, so that A (B C) = (B D) (B C). From (e)
[ \ [ [ \ [
we get the equality
(B D) (B C)=((B D) B) ((B D) C).
[ \ [ [ \ [ [ \
Since (B D) B =B, the equality becomes
[ \
(B D) (B C)=B ((B D) C),
[ \ [ [ [ \
which is B (D C). Hence (B D) (B C) = B (D C) for any sets
[ \ [ \ [ [ \
B, C and D. This is the equality (f) above. The point to all this is, however,
that once we have an equality such as (e) that holds for any sets A, B, and C,
we can get other equalities by taking A, B, and C to be any particular sets we
wish.
PROBLEMS
1. List all the subsets of the set a .
f g
2. List all the subsets of the set a;b .
f g
3. List all the subsets of the set a;b;c .
f g
4. List all the subsets of the set .
;
5. List all the subsets of the set .
f;g
6. List all the subsets of the set a;b; a;b
f f gg
7. List the elements in n Z:mn=100 for some m Z .
f 2 2 g
2
8. List the elements in n Z:n n<211 .
f 2 (cid:0) g
9. Prove directly that A (B C)=(A B) (A C).
[ \ [ \ [
10. Prove that A ((A B) C)=A (B C).
[ [ \ [ \
11. Prove that (A B) (B A)= .
n \ n ;
12. Prove that A B = if and only if A B.
n ; (cid:26)