Algebra(SergeLang) - CSDN文库

5星 · 超过95%的资源需积分: 48 577 浏览量更新于2023-05-21 评论 12 收藏 10.66MB PDF 举报

身份认证购VIP最低享 7 折!

领优惠券(最高得80元）

资源详情

资源评论

资源推荐

Graduate

Texts

in

Mathematics

211

Editorial

Board

S.

Axler

F.W.

Gehring

K.A.

Ribet

Springer

New

York

Berlin

Heidelberg

Barcelona

Hong

Kong

London

Milan

Paris

Singapore

Tokyo

vi

FOREWORD

I

have

inserted

cross-references

to

help

them

do

this,

but

different

people

will

make

different

choices

at

different

times

depending

on

different

circumstances.

The

book

splits

naturally

into

several

parts.

The

first

part

introduces

the

basic

notions

of

algebra.

After

these

basic

notions,

the

book

splits

in

two

major

directions:

the

direction

of

algebraic

equations

including

the

Galois

theory

in

Part

II;

and

the

direction

of

linear

and

multilinear

algebra

in

Parts

III

and

IV.

There

is

some

sporadic

feedback

between

them,

but

their

unification

takes

place

at

the

next

level

of

mathematics,

which

is

suggested,

for

instance,

in

15

of

Chapter

VI.

Indeed,

the

study

of

algebraic

extensions

of

the

rationals

can

be

carried

out

from

two

points

of

view

which

are

complementary

and

interrelated:

representing

the

Galois

group

of

the

algebraic

closure

in

groups

of

matrices

(the

linear

approach),

and

giving

an

explicit

determination

of

the

irrationalities

gen-

erating

algebraic

extensions

(the

equations

approach).

At

the

moment,

repre-

sentations

in

GL

2

are

at

the

center

of

attention

from

various

quarters,

and

readers

will

see

GL

2

appear

several

times

throughout

the

book.

For

instance,

I

have

found

it

appropriate

to

add

a

section

describing

all

irreducible

characters

of

GL

2

(F)

when

F

is

a

finite

field.

Ultimately,

GL

2

will

appear

as

the

simplest

but

typical

case

of

groups

of

Lie

types,

occurring

both

in

a

differential

context

and

over

finite

fields

or

more

general

arithmetic

rings

for

arithmetic

applications.

After

almost

a

decade

since

the

second

edition,

I

find

that

the

basic

topics

of

algebra

have

become

stable,

with

one

exception.

I

have

added

two

sections

on

elimination

theory,

complementing

the

existing

section

on

the

resultant.

Algebraic

geometry

having

progressed

in

many

ways,

it

is

now

sometimes

return-

ing

to

older

and

harder

problems,

such

as

searching

for

the

effective

construction

of

polynomials

vanishing

on

certain

algebraic

sets,

and

the

older

elimination

procedures

of

last

century

serve

as

an

introduction

to

those

problems.

Except

for

this

addition,

the

main

topics

of

the

book

are

unchanged

from

the

second

edition,

but

I

have

tried

to

improve

the

book

in

several

ways.

First,

some

topics

have

been

reordered.

I

was

informed

by

readers

and

review-

ers

of

the

tension

existing

between

having

a

textbook

usable

for

relatively

inex-

perienced

students,

and

a

reference

book

where

results

could

easily

be

found

in

a

systematic

arrangement.

I

have

tried

to

reduce

this

tension

by

moving

all

the

homological

algebra

to

a

fourth

part,

and

by

integrating

the

commutative

algebra

with

the

chapter

on

algebraic

sets

and

elimination

theory,

thus

giving

an

intro-

duction

to

different

points

of

view

leading

toward

algebraic

geometry.

The

book

as

a

text

and

a

reference

In

teaching

the

course,

one

might

wish

to

push

into

the

study

of

algebraic

equations

through

Part

II,

or

one

may

choose

to

go

first

into

the

linear

algebra

of

Parts

III

and

IV.

One

semester

could

be

devoted

to

each,

for

instance.

The

chapters

have

been

so

written

as

to

allow

maximal

flexibility

in

this

respect,

and

I

have

frequently

committed

the

crime

of

lese-

Bourbaki

by

repeating

short

argu-

ments

or

definitions

to

make

certain

sections

or

chapters

logically

independent

of

each

other.

FOREWORD

vii

Granting

the

material

which

under

no

circumstances

can

be

omitted

from

a

basic

course,

there

exist

several

options

for

leading

the

course

in

various

direc-

tions.

It

is

impossible

to

treat

all

of

them

with

the

same

degree

of

thoroughness.

The

precise

point

at

which

one

is

willing

to

stop

in

any

given

direction

will

depend

on

time,

place,

and

mood.

However,

any

book

with

the

aims

of

the

present

one

must

include

a

choice

of

topics,

pushing

ahead

in

deeper

waters,

while

stopping

short

of

full

involvement.

There

can

be

no

universal

agreement

on

these

matters,

not

even

between

the

author

and

himself.

Thus

the

concrete

decisions

as

to

what

to

include

and

what

not

to

incl

ude

are

finally

taken

on

grounds

of

general

coherence

and

aesthetic

balance.

Anyone

teaching

the

course

will

want,to

impress

their

own

personality

on

the

material,

and

may

push

certain

topics

with

more

vigor

than

I

have,

at

the

expense

of

others.

Nothing

in

the

present

book

is

meant

to

inhibit

this.

Unfortunately,

the

goal

to

present

a

fairly

comprehensive

perspective

on

algebra

required

a

substantial

increase

in

size

from

the

first

to

the

second

edition,

and

a

moderate

increase

in

this

third

edition.

These

increases

require

some

decisions

as

to

what

to

omit

in

a

given

course.

Many

shortcuts

can

be

taken

in

the

presentation

of

the

topics,

which

admits

many

variations.

For

instance,

one

can

proceed

into

field

theory

and

Galois

theory

immediately

after

giving

the

basic

definitions

for

groups,

rings,

fields,

polynomials

in

one

variable,

an

vector

spaces.

Since

the

Galois

theory

gives

very

quickly

an

impression

of

dpth,

this

is

very

satisfactory

in

many

respects.

It

is

appropriate

here

to

recall

my

original

indebtedness

to

Artin,

who

first

taught

me

algebra.

The

treatment

of

the

basics

of

Galois

theory

is

much

influenced

by

the

presentation

in

his

own

monograph.

Audience

and

background

As

I

already

stated

in

the

forewords

of

previous

editions,

the

present

book

is

meant

for

the

graduate

level,

and

I

expect

most

of

those

coming

to

it

to

have

had

suitable

exposure

to

some

algebra

in

an

undergraduate

course,

or

to

have

appropriate

mathematical

maturity.

I

expect

students

taking

a

graduate

course

to

have

had

some

exposure

to

vector

spaces,

linear

maps,

matrices,

and

they

will

no

doubt

have

seen

polynomials

at

the

very

least

in

calculus

courses.

My

books

Undergraduate

Algebra

and

Linear

Algebra

provide

more

than

enough

background

for

a

graduate

course.

Such

elementary

texts

bring

out

in

parallel

the

two

basic

aspects

of

algebra,

and

are

organized

differently

from

the

present

book,

where

both

aspects

are

deepened.

Of

course,

some

aspects

of

the

linear

algebra

in

Part

III

of

the

present

book

are

more

"elementary"

than

some

aspects

of

Part

II,

which

deals

with

Galois

theory

and

the

theory

of

polynomial

equations

in

several

variables.

Because

Part

II

has

gone

deeper

into

the

study

of

algebraic

equations,

of

necessity

the

parallel

linear

algebra

occurs

only

later

in

the

total

ordering

of

the

book.

Readers

should

view

both

parts

as

running

simultaneously.

剩余932页未读，继续阅读

评论1

dengxiatingyu

2023-01-28

可以下载，2002版，只是没有封面

qq_41948503

粉丝: 2
资源: 15

会员权益专享

图片转文字

全年可省5，000元立即开通

最新资源

资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！点击此处反馈