Algebra
内容简介
Book Description
"Lang's Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books." NOTICES OF THE AMS "The author has an impressive knack for presenting the important and interesting ideas of algebra in just the right way, and he never gets bogged down in the dry formalism which pervades some parts of algebra." MATHEMATICAL REVIEWS This book is intended as a basic text for a one-year course in algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra. For the revised third edition, the author has added exercises and made numerous corrections to the text.
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作者简介
Serge Lang (May 19, 1927–September 12, 2005) was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He was a member of the Bourbaki group.
He was born in Paris in 1927, and moved with his family to California as a teenager. He graduated from CalTech in 1946, and received a doctorate from Princeton University in 1951. He had positions at the University of Chicago and Columbia University (from 1955, leaving 1971 in a dispute). At the time of his death he was professor emeritus of mathematics at Yale University.
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目录
Part One The Basic Objects of Algebra
Chapter 1 Groups
1. Monoids
2. Groups
3. Normal subgroups
4. Cyclic groups
5. Operations of a group on a set
6. Sylow subgroups
7. Direct sums and free abelian groups
8. Finitely generated abelian groups
9. The dual group
10. Inverse limit and completion
11. Categories and functors
12. Free groups
Chapter 2 Rings
1. Rings and homomorphisms
2. Commutative rings
3. Polynomials and group rings
4. Localization
5. Principal and factorial rings
Chapter 3 Modules
Chapter 4 Polynomlals
Part Two Algebraic Equations
Chapter 5 Algebralc Extensions
Chapter 6 Galois Theory
Chapter 7 Extensions of Rings
Chapter 8 Transcendental Extensions
Chapter 9 Algebraic Spaces
Chapter 10 Noetherial Rings and Modules
Chapter 11 Real Fields
Chapter 12 Absolute Values
Part Three Liear Alebar and Reqresentations
Chapter 13 Matrices and Linear Maps
Chapter 14 Representatlon of One Endomorphism
Chapter 15 Structure of Bilinear Forms
Chapter 16 The Tensor Product
Chapter 17 Smisimplicity
Chapter 18 Representations of Finite Groups
Chapter 19 The Alternating Product
Part Four Homological Algebra
Chapter 20 General Homology Theory
Chapter 21 Finite Free Resolutions
Appendix 1 The Transcendence of e and
Appendix 2 Some Set Theory
Bibliography
Index
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读书文摘
We may as well be open about ir and admit that yes,..... Shame on us!
Joke 1.1. Definition: A group is a groupoid with a single object.
_Proof._ … Let [;a_i;] be …
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