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《代数学基础》由科学出版社出版。
作者简介
作者:（俄）沙法列维奇（Igor R. Shafarevich）
目录
Preface
1.What is Algebra？
2.Fields
3.Commutative Rings
4.Homomorphisms and Ideals
5.Modules
6.Algebraic Aspects of Dimension
7.The Algebraic View of Infinitesimal Notions
8.Noncommutative Rings
9.Modules over Noncommutative Rings
10.Semisimple Modules and Rings
11.Division Algebras of Finite Rank
12.The Notion of a Group
13.Examples of Groups： Finite Groups
14.Examples of Groups： Infinite Discrete Groups
15.Examples of Groups： Lie Groups and Algebraic Groups
16.General Results of Group Theory
17.Group Representations
A.Representations of Finite Groups
B.Representations of Compact Lie Groups
18.Some Applications of Groups
A.Galois Theory
B.The Galois Theory of Linear Differential Equations （Picard Vessiot Theory）
C.Classification of Unramified Covers
D.Invariant Theory
E.Group Representations and the Classification of Elementary Particles
19.Lie Algebras and Nonassociative Algebra
A.Lie Algebras
B.Lie Theory
C.Applications of Lie Algebras
D.Other Nonassociative Algebras
20.Categories
21.Homological Algebra
A.Topological Origins of the Notions of Homological Algebra
B.Cohomology of Modules and Groups
C.Sheaf Cohomology
22.K—theory
A.Topological K—theory
B.Algebraic K—theory
Comments on the Literature
References
Index of Names
Subject Index
文摘
版权页：
插图：
The question of the existence of central division algebras of finite rank over a given field K and of their structure depends in a very essential way on particular properties of K.We have already met one very simple result in this direction：over an algebraically closed field K there do not exist division algebras of finite rank other than K.In particular this holds for the complex number field.
For the case of the real number field the situation is not much more complicated.
Ⅰ.Frobenius' Theorem.The only division algebras of finite rank over the real number field R are R itself， the complex number field C，and the quaterntion algebra H.
Here are another two cases when the situation is simple.
Ⅱ.Wedderburn's Theorem.Over a Finite field K， there do not exist any central division algebras of finite rank other than K itself.
In other words，a finite division algebra is commutative.This is of course interesting for projective geometry， since it shows that for firtite projective geometries of dimension>2， Pappus' theorem follows from the other axioms （and in dimension 2，from Desargues' theorem）.
《代数学基础》由科学出版社出版。
作者简介
作者:（俄）沙法列维奇（Igor R. Shafarevich）
目录
Preface
1.What is Algebra？
2.Fields
3.Commutative Rings
4.Homomorphisms and Ideals
5.Modules
6.Algebraic Aspects of Dimension
7.The Algebraic View of Infinitesimal Notions
8.Noncommutative Rings
9.Modules over Noncommutative Rings
10.Semisimple Modules and Rings
11.Division Algebras of Finite Rank
12.The Notion of a Group
13.Examples of Groups： Finite Groups
14.Examples of Groups： Infinite Discrete Groups
15.Examples of Groups： Lie Groups and Algebraic Groups
16.General Results of Group Theory
17.Group Representations
A.Representations of Finite Groups
B.Representations of Compact Lie Groups
18.Some Applications of Groups
A.Galois Theory
B.The Galois Theory of Linear Differential Equations （Picard Vessiot Theory）
C.Classification of Unramified Covers
D.Invariant Theory
E.Group Representations and the Classification of Elementary Particles
19.Lie Algebras and Nonassociative Algebra
A.Lie Algebras
B.Lie Theory
C.Applications of Lie Algebras
D.Other Nonassociative Algebras
20.Categories
21.Homological Algebra
A.Topological Origins of the Notions of Homological Algebra
B.Cohomology of Modules and Groups
C.Sheaf Cohomology
22.K—theory
A.Topological K—theory
B.Algebraic K—theory
Comments on the Literature
References
Index of Names
Subject Index
文摘
版权页：
插图：
The question of the existence of central division algebras of finite rank over a given field K and of their structure depends in a very essential way on particular properties of K.We have already met one very simple result in this direction：over an algebraically closed field K there do not exist division algebras of finite rank other than K.In particular this holds for the complex number field.
For the case of the real number field the situation is not much more complicated.
Ⅰ.Frobenius' Theorem.The only division algebras of finite rank over the real number field R are R itself， the complex number field C，and the quaterntion algebra H.
Here are another two cases when the situation is simple.
Ⅱ.Wedderburn's Theorem.Over a Finite field K， there do not exist any central division algebras of finite rank other than K itself.
In other words，a finite division algebra is commutative.This is of course interesting for projective geometry， since it shows that for firtite projective geometries of dimension>2， Pappus' theorem follows from the other axioms （and in dimension 2，from Desargues' theorem）.
| ISBN | 7030166914,9787030166913 |
|---|---|
| 出版社 | 科学出版社 |
| 作者 | 沙法列维奇 (Igor R.Shafarevich) |
| 尺寸 | 5 |