Advanced Modern Algebra.pdf

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Advanced Modern Algebra
by Joseph J. Rotman
Hardcover:
1040 pages
Publisher:
Prentice Hall; 1st edition (2002); 2nd printing (2003)
Language:
English
ISBN:
0130878685
Book Description
This book's organizing principle is the interplay between groups and rings,
where “rings” includes the ideas of modules. It contains basic definitions,
complete and clear theorems (the first with brief sketches of proofs), and
gives attention to the topics of algebraic geometry, computers, homology,
and representations. More than merely a succession of definition-theorem-proofs,
this text put results and ideas in context so that students can appreciate why
a certain topic is being studied, and where definitions originate. Chapter
topics include groups; commutative rings; modules; principal ideal domains;
algebras; cohomology and representations; and homological algebra. For
individuals interested in a self-study guide to learning advanced algebra and
its related topics.
Book Info
Contains basic definitions, complete and clear theorems, and gives attention
to the topics of algebraic geometry, computers, homology, and representations.
For individuals interested in a self-study guide to learning advanced algebra
and its related topics.
To my wife
Marganit
and our two wonderful kids,
Danny and Ella,
whom I love very much
Contents
Second Printing .
Preface . . . . .
Etymology . . .
Special Notation
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viii
ix
xii
xiii
1
1
15
25
39
39
40
51
62
73
82
96
Chapter 1 Things Past
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1. Some Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3. Some Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 2
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
Groups I
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Introduction . . . . .
Permutations . . . . .
Groups . . . . . . . .
Lagrange’s Theorem
Homomorphisms . .
Quotient Groups . . .
Group Actions . . . .
Chapter 3
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
Commutative Rings I
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116
116
126
131
143
151
158
159
171
182
Introduction . . . . . . . . . . .
First Properties . . . . . . . . .
Polynomials . . . . . . . . . . .
Greatest Common Divisors . . .
Homomorphisms . . . . . . . .
Euclidean Rings . . . . . . . . .
Linear Algebra . . . . . . . . .
Vector Spaces . . . . . . . . .
Linear Transformations . . . .
3.8. Quotient Rings and Finite Fields
v

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