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Algebra (Serge Lang)
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在学习代数学之余，值得一看的代数学书籍。里面介绍了更为丰富的代数学概念和结论。
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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
Serge
Lang
Department
of
Mathematics
Yale
University
New
Haven,
CT
96520
USA
Editorial
Board
S.
Axler
Mathematics
Department
San
Francisco
State
University
San
Francisco,
CA
94132
USA
F.
W.
Gehring
Mathematics
Department
East
Hall
University
of
Michigan
Ann
Arbor,
MI
48109
USA
K.A.
Ribet
Mathematics
Department
University
of
California
at
Berkeley
Berkeley,
CA
947203840
USA
Mathematics
Subject
Classification
(2000):
1301,
1501,
1601,
2001
Library
of
Congress
CataloginginPublication
Data
Algebra
I
Serge
Lang.Rev.
3rd
ed.
p.
em.

(Graduate
texts
in
mathematics;
211)
Includes
bibliographical
references
and
index.
ISBN
038795385X
(alk.
paper)
1.
Algebra.
I.
Title.
II.
Series.
QA
154.3.L3
2002
512dc21
2001054916
Printed
on
acidfree
paper.
This
title
was
previously
published
by
AddisonWesley,
Reading,
MA
1993.
<9
2002
SpringerVerlag
New
York,
Inc.
All
rights
reserved.
This
work
may
not
be
translated
or
copied
in
whole
or
in
part
without
the
written
pennission
of
the
publisher
(SpringerVerlag
New
York,
Inc.,
175
Fifth
Avenue,
New
York,
NY
10010,
USA),
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in
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or
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analysis.
Use
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names,
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9
8
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6
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432
1
ISBN
0387
95385X
SPIN
10855619
Springer
Verlag
New
York
Berlin
Heidelberg
A
member
of
Be
rtelsmannSp
ringer
Science+Business
Media
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FOREWORD
The
present
book
is
meant
as
a
basic
text
for
a
oneyear
course
in
algebra,
at
the
graduate
level.
A
perspective
on
algebra
As
I
see
it,
the
graduate
course
in
algebra
must
primarily
prepare
students
to
handle
the
algebra
which
they
will
meet
in
all
of
mathematics:
topology,
partial
differential
equations,
differential
geometry,
algebraic
geometry,
analysis,
and
representation
theory,
not
to
speak
of
algebra
itself
and
algebraic
number
theory
with
all
its
ramifications.
Hence
I
have
inserted
throughout
references
to
papers
and
books
which
have
appeared
during
the
last
decades,
to
indicate
some
of
the
directions
in
which
the
algebraic
foundations
provided
by
this
book
are
used;
I
have
accompanied
these
references
with
some
motivating
comments,
to
explain
how
the
topics
of
the
present
book
fit
into
the
mathematics
that
is
to
come
subsequently
in
various
fields;
and
I
have
also
mentioned
some
unsolved
problems
of
mathematics
in
algebra
and
number
theory.
The
abc
conjecture
is
perhaps
the
most
spectacular
of
these.
Often
when
such
comments
and
examples
occur
out
of
the
logical
order,
especially
with
examples
from
other
branches
of
mathematics,
of
necessity
some
terms
may
not
be
defined,
or
may
be
defined
only
later
in
the
book.
I
have
tried
to
help
the
reader
not
only
by
making
crossreferences
within
the
book,
but
also
by
referring
to
other
books
or
papers
which
I
mention
explicitly.
I
have
also
added
a
number
of
exercises.
On
the
whole,
I
have
tried
to
make
the
exercises
complement
the
examples,
and
to
give
them
aesthetic
appeal.
I
have
tried
to
use
the
exercises
also
to
drive
readers
toward
variations
and
appli
cations
of
the
main
text,
as
well
as
toward
working
out
special
cases,
and
as
openings
toward
applications
beyond
this
book.
Organization
Unfortunately,
a
book
must
be
projected
in
a
totally
ordered
way
on
the
page
axis,
but
that's
not
the
way
mathematics
"is",
so
readers
have
to
make
choices
how
to
reset
certain
topics
in
parallel
for
themselves,
rather
than
in
succession.
v
vi
FOREWORD
I
have
inserted
crossreferences
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.
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