herstein abstract algebra, Notas de estudo de Matemática

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i. n. herstein

University of Chicago

TOPICS

IN

ALGEBRA

2nd

edition

JOHN WILEY &

SONS

New York • Chichester • Brisbane • Toronto

• Singapore

Preface to the Second Edition

I approached rev1smg Topics in Algebra with a certain amount of

trepidation. On the whole, I was satisfied with the first edition and did

not want to tamper with it. However, there were certain changes I felt

should be made, changes which would not affect the general style or

content, but which would make the book a little more complete. I

hope that I have achieved this objective in the present version.

For the most part, the major changes take place in the chapt¥r on

group theory. When the first edition was written it was fairly un-

common for a student learning abstract algebra to have had any

previous exposure to linear algebra. Nowadays quite the opposite is

true; many students, perhaps even a majority, have learned something

about 2 x 2 matrices at this stage. Thus I felt free here to draw on

2 x 2 matrices for examples and problems. These parts, which

depend on some knowledge of linear algebra, are indicated with a #.

In the chapter on groups I have largely expanded one section, that

on Sylow's theorem, and added two others, one on direct products and

one on the structure of finite abelian groups.

In the previous treatment of Sylow's theorem, only the existence of a

Sylow subgroup was shown. This was done following the proof of

Wielandt. The conjugacy of the Sylow subgroups and their number

were developed in a series of exercises, but not in the text proper.

Now all the parts of Sylow's theorem are done in the text materi9-l.

iii

iv Preface to the Second Edition

In addition to the proofpreviously given for the existence, two other

proofs of existence are carried out. One could accuse me ofoverkill

atthispoint, probably rightfully so. Thefactofthe matteris thatSylow's

theoremis important, thateachproof illustrates a different aspect ofgroup

theoryand, aboveall,that I love Sylow's theorem. Theproof ofthecon-

jugacy and number ofSylow subgroups exploitsdouble cosets. A by-product

ofthisdevelopment isthat ameansis given for finding Sylow subgroups ina

largesetof symmetricgroups.

Forsome mysterious reason known only to myself, I hadomitted direct

productsin thefirst edition. Whyis beyond me. Thematerial is easy,

straightforward, andimportant. This lacunaisnow filled in thesection

treating direct products. Withthisin hand,I go on in the next section to

provethe decomposition ofa finite abelian group as a direct product of

cyclicgroups andalso provetheuniqueness ofthe invariantsassociated with

this decomposition. In pointoffact, this decomposition wasalready in the

first edition, atthe end ofthe chapter onvectorspaces, as a consequence of

the structure offinitely generated modules over Euclidean rings. However,

thecase ofa finitegroup isofgreat importance by itself;thesection onfinite

abelian groups underlines thisimportance. Its presencein the chapteron

groups,an early chapter, makesitmore likelythat itwillbe taught.

One other entiresection has been addedatthe endofthe chapter onfield

theory. I feltthatthe student should seean explicitpolynomial over an

explicit field whose Galois groupwas thesymmetric groupof degree 5, hence

one whose roots could not be expressed byradicals. Inorder to do so, a

theorem is first proved which gives a criterion that anirreducible poly-

nomial ofdegree p, p a prime, over the rational fieldhave SP as its Galois

group. Asan application ofthis criterion,an irreducible polynomial of

degree 5 is given, over the rational field, whose Galois groupis the symmetric

group ofdegree 5.

There areseveralother additions. More than 150 new problems areto be

found here. They are of varying degreesofdifficulty. Many are routine

and computational, many are very djfficult. Furthermore,

some inter-

polatory remarks are made about problems that have given readers a great

dealof difficulty. Some paragraphs have been inserted, others rewritten,at

places where the writing hadpreviously been obscure or too terse.

Above I have described whatI have added. What gave me greater

difficulty about the revision was, perhaps, that which I have not added. I

debated for a long time with myself whether or not to adda chapter on

category theory and some elementary functors, whether or not to enlarge the

material on modules substantially. After a great deal of thought and soul-

searching, I decided not to do so. The book, as stands, has a certain concrete-

ness about it with which this new material would not blend. It could be

made to blend, but this would require a complete reworking of the material

Preface to the First Edition

The idea to write this book, and more important the desire to do so, is

a direct outgrowth of a course I gave in the academic year 1959-1960 at

Cornell University. The class taking this course consisted, in large part,

of the most gifted sophomores in mathematics at Cornell. It was my

desire to experiment by presenting to them material a little beyond that

which is usually taught in algebra at the junior-senior level.

I have aimed this book to be, both in content and degree of sophisti-

cation, about halfway between two great classics, A Survey of M~dern

Algebra, by Birkhoff and MacLane, and Modern Algebra, by Van der

Waerden.

The last few years have seen marked changes in the instruction given

in mathematics at the American universities. This change is most

notable at the upper undergraduate and beginning graduate levels.

Topics that a few years ago were considered proper subject matter for

semiadvanced graduate courses in algebra have filtered down to, and

are being taught in, the very first course in abstract algebra. Convinced

that this filtration will continue and will become intensified in the next

few years, I have put into this book, which is designed to be used as the

student's first introduction to algebra, material which hitherto has been

considered a little advanced for that stage of the game.

There is always a great danger when treating abstract ideas to intro-

duce them too suddenly and without a sufficient base of examples to

render them credible or natural. In order to try to mitigate this, I have

tried to motivate the concepts beforehand and to illustrate them in con-

crete situations. One of the most telling proofs of the worth of an abstract

vii

Contents

1 Preliminary Notions

Set Theory

1.2 Mappings

TheIntegers

2 Group Theory

Definitionofa Group

Some

ExamplesofGroups

Some

Preliminary Lemmas

Subgroups

A Counting

Principle

Normal Subgroups

and Quotient

Groups

Homomorphisms

Automorphisms

Cayley's Theorem

Permutation

Groups

Another

Counting Principle

Sylow's Theorem

Direct

Products

Finite Abelian

Groups

ix

2

26

27

29

37

66

75

82

X Contents

3 Ring Theory

Definition andExamples ofRings

Some Special Classes ofRings

Homomorphisms

Ideals andQuotient Rings

More IdealsandQuotient Rings

3.6 The

FieldofQuotients ofanIntegral

Domain 140

Euclidean Rings

3.8 A

Particular Euclidean Ring

3.9 Polynomial

Rings

Polynomials over the RationalField

Polynomial Rings over Commutative

Rings 161

4 Vector Spaces and Modules

Elementary Basic Concepts

4.2 Linear Independence

andBases

4.3 Dual Spaces

4.4 Inner Product

Spaces

4.5 Modules

5 Fields

5.1 Extension Fields

5.2 The

Transcendence

of e

RootsofPolynomials

Construction with Straightedgeand Compass

5.5 More

About Roots

TheElementsof Galois Theory

Solvability by Radicals

Galois Groups over the Rationals

6 Linear Transformations

6.1 The

Algebra of Linear Transformations

6.2 Characteristic

Roots

Matrices

6.4 Canonical Forms: Triangular Form

1

Prelilllinary Notions

One of the amazing features of twentieth century mathematics has

been its recognition of the power of the abstract approach. This has

given rise to a large body of new results and problems and has, in fact,

led us to open up whole new areas of mathematics whose very existence

had not even been suspected.

In the wake of these developments has come not only a new

mathematics but a fresh outlook, and along with this, simple new

proofs of difficult classical results. The isolation of a problem inl'o its

basic essentials has often revealed for us the proper setting, in the whole

scheme of things, of results considered to have been special and apart

and has shown us interrelations between areas previously thought to

have been unconnected.

The algebra which has evolved as an outgrowth of all this is not

only a subject with an independent life and vigor-it is one of the

important current research areas in mathematics-but it also serves as

the unifying thread which interlaces almost all of mathematics-

geometry, number theory, analysis, topology, and even applied

mathematics.

This book is intended as an introduction to that part of mathematics

that today goes by the name of abstract algebra. The term "abstract"

is a highly subjective one; what is abstract to one person is very often

concrete and down-to-earth to another, and vice versa. In relation to

the current research activity in algebra, it could be described as

"not too abstract"; from the point of view of someone schooled in the

1

Sec.1.1 Set Theory

and if A is the subset ofpositive integers,

thenwe can describe A as

A = {a E S I a > 0}. Anotherexampleof

this:If Sis theset consistingof

theobjects (1), (2),..., (10), then thesubset A

consistingof(1), (4), (7),

(10) could bedescribedby A = {(i) E S I i =3n + 1, n = 0, 1, 2, 3}.

Given two sets wecancombinethemto form

newsets. Thereisnothing

sacredorparticularaboutthisnumbertwo; wecan

carry out the samepro-

cedureforany numberofsets, finiteorinfinite,andin

fact we shall. We

doso for two first becauseitillustratesthegeneralconstruction

butisnot

obscuredby the additional notational difficulties.

DEFINITION The union ofthetwo sets A and B, writtenas A u B, isthe

set {x I x E A or

x E B}.

Awordabout theuseof "or." InordinaryEnglishwhenwe saythat

something is one

orthe otherwe.implythatitisnot both. Themathematical

"or" is quite different,

atleast when we arespeakingaboutset theory. For

when we say that x is in Aorx is in B

wemean x is in at leastone of AorB,and

maybein both.

Letus consider a few examplesof theunion

oftwo sets. For anyset A,

A u A = A; infact, whenever B is a subsetof A,

A u B = A. If A isthe

set{x 1

, x

2

, x

3

} (i.e.,theset whose elementsarex

1

, x

2

, x

3

)

andif B istheset

{y

1

,y

2

, xd, then A u B = {x

1

, x

2

, x

3

,y

1

,y

2

}. If A isthesetof

all blonde-

hairedpeopleand if B isthesetofall people who smoke,then A

u B

consistsofallthe people who either have blonde hair orsmokeor both.

Pictorially we can illustrate the unionofthe two sets A and B by

Here, A is the circle on the left, B that on the right,and A u B is the shaded

part.

DEFINITION The intersection of the two sets A and B, written

as A r. B,

is the set {x I x E A and x

E B}.

The intersection of A and B is thus the set of all elements which are both

in A and in B. In analogy with the examples used to illustrate the union of

two sets, let us see what

the intersections are in those very examples. For

4 Preliminary Notions Ch. 1

anyset A,A n

A = A; infact,if B isanysubsetof A, then A n B = B.

If A istheset

{x

1

, x

2

, x

3

}and B

theset {y

1

,y

2

, xd,

then A n B = {xd

(wearesupposing no y isan x). If

A isthesetofallblonde-haired

people

andif B isthe

setofall peoplethat

smoke,then A

n B isthesetofall

blonde-hairedpeople whosmoke. Pictorially we canillustrate theinter-

sectionofthe

two sets A and B

by

Here A is

thecircleon theleft,

B thaton theright,

whiletheirintersection

isthe shaded part.

Two setsaresaid tobe disjoint iftheirintersection is empty,that is, is

the null set. Forinstance,

if A isthesetofpositive

integersand B the

setof

negative integers,then A andBaredisjoint. Note howeverthat if Cisthe

set ofnonnegative integers andif D isthe setofnonpositive integers, then

they are notdisjoint, for theirintersection consistsofthe integer 0, andso is

not empty.

Before we generalize unionandintersection from two sets to anarbitrary

number ofthem, we should like to prove a little proposition interrelating

unionandintersection. Thisisthe firstofa whole host ofsuch resqlts that

can be proved; someofthesecan be found in the problems atthe end ofthis

section.

PROPOSITION

For any three sets,A, B, C we have

A

n (B u C) = (A

n B) u (A n C).

Proof The proof will consist ofshowing, to begin with, the relation

(A

n B) u (A n C) c A n (B u C) and then the converse relation

A n (B u C) c

(A n B) u (A n C).

We first dispose of (A n B)

u (A n C) c A n

(B u C). Because

B c B u C, it is immediate that A n B c A n (B u C). Ina similar

manner, A n C c A n (B u C). Therefore

(A n B) u (A n C) c (A n (B u C)) u (A n (B u

C)) = A n (B u C).

Now for the other direction. Given an element x E A n (B u C),

first of all it must

be an element of A.

Secondly, as an element

in B u C it

is

either in B or in C.

Suppose the former; then as an element

both of A and

of B, x must be in A n B. The second possibility, namely, x E C, leads us

6 Preliminary Notions Ch. 1

A few remarksabout theCartesian product. Giventhe two sets A and B

we could construct thesets A x B and B x A fromthem. As sets theseare

distinct,yetwe feel thatthey must be closely related. Given three sets A,

B,C we canconstructmany Cartesian products fromthem:for instance, the

set A

x D, where D =

B x C; theset E

x C, where E=A

x B; and

alsothe setofallordered

triples (a, b, c) where

a E A, bE B, and

c E C.

Thesegive us threedistinct sets, yet here, also, we feel thatthese sets must

beclosely related. Of course, wecancontinue this process with more and

moresets. Toseethe exact relation betweenthem we shall have to wait

untilthe nextsection,where we discuss one-to-one correspondences.

Given anyindex set T we could definethe Cartesian productofthe sets

Aa as

ex varies over T; since we

shallnot need so general aproduct, we do

not botherto define it.

Finally, wecanconsider the Cartesianproduct ofa set A with itself,

A x

A. Notethat ifthe

set A is a finite set

having n elements, then theset

A x A is also a finite set,

buthas n

2

elements. Thesetofelements

(a,a) in

A x A is called the

diagonal of A x A.

A subset R of

A x A is said to define

an equivalencerelation

on A if

1. (a, a) E

R for all a EA.

2. (a, b) E

R implies (b, a) E R.

3. (a, b)

ERand (b, c) E R imply

that (a,c) E R.

Instead ofspeakingaboutsubsets of A x A we canspeakaboutabinary

relation (one between two elements

of A) on A itself, defining

b to be related

to a

if (a, b) E R. Theproperties

1, 2, 3ofthe subset R immediately-translate

into theproperties 1, 2, 3 ofthedefinition below.

DEFINITION The binary relation "'

on A is said to

be an equivalence

relation on A iffor

all a, b,c in A

I. a "' a.

2. a "' b implies

b "' a.

3. a "'

b and b "' c imply

a "' c.

Thefirst ofthese properties is

called riflexivity, the

second, symmetry,

and

the third,

transitivity.

Theconcept of an equivalence relation isan extremely important one

andplays a central role in all ofmathematics. We illustrate it with a few

examples.

Example 1.1.1 Let S be any set and define a "' b, for a, b E S, if and

only if a = b. This clearly defines an equivalence relation on S. In fact, an

equivalence relation is a generalization of equality, measuring equality up

to some property.

Sec. 1.1 Set Theory

Example

1 .1.2 Let S be thesetofall integers.

Given a,b E S, define

a "' b if a -

b isaneven integer. Weverify

thatthis definesanequivalence

relation

of S.

I. Since 0 = a a is even,

a IV a.

2. If a "' b, thatis,if

a - b is even,then b - a =

-(a- b) is also even,

whence

b "' a.

3. If a "' b and b

IV c, then both a - b and

b - c are even, whence

a - c = (a - b)

• (b - c) isalso even, proving

that a "' c.

Example 1.1.3 Let

S be thesetofall integersand

let n > 1 be a fixed

integer. Define for a, bE S, a

"' b if a - b is a multipleof

n. Weleaveit

asanexercise to provethat

this definesanequivalence relation

on S.

Example 1 .1.4 Let S be the

set ofall triangles in the

plane. Two

triangles

aredefined tobeequivalent

iftheyaresimilar (i.e., have corre-

sponding

angles equal). This defines

anequivalence relationon S.

Example 1.1.

Let S be thesetofpoints

in theplane. Twopoints a and

b aredefined to

beequivalentifthey are equidistant

fromtheorigin. A

simple check verifies

thatthis definesanequivalence

relation on S.

There are many more

equivalence relations; we shall

encounter a few as

we proceed in the book.

DEFINITION

If A is a setand

if"' isanequivalence relation

on A, then

the equivalence

class of a E A istheset

{x E A I a "' x}. We write it as cl(a).

Inthe

examplesjustdiscussed, what

are the equivalence classes?

In

Example

1.1.1, the equivalence class

of a consists merelyof a

itself. In

Example

1.1.2 the equivalence class

of a consistsofall the integers

ofthe

form a + 2m,

where m = 0, ±1,±2,...;

in this example there are only

two distinct equivalence classes, namely,

cl(O) and cl(l). InExample 1.1.3,

the equivalence class

of a consistsofall integers

of the form a + kn where

k = 0, ± I, ±

2,...; here there are n distinct equivalence classes,

namely

cl(O),cl(l),

... ,cl(n- 1). In Example

1.1.5, the equivalence class

of a

consists

of all the points in the plane

which lie on the circle which

has its

center at the origin and passes

through a.

Although

we have made quite a few definitions,

introduced some concepts,

and have even established a simple little proposition,

one

could say in all

fairness that up to this point

we have not proved any result

of real substance.

We are now about

to prove the first genuine

result in the book. The proof

of this theorem is not very difficult-actually

it is quite easy-but

nonetheless

the result

it embodies will be of great use

to us.

7