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Algebra
Basic Concepts of Abstract Algebra
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Textbooks for Tomorrow's Scientists
Textbooks for Tomorrow’s Scientists introduces advanced textbooks on a wide range of topics in STM subject areas. These books offer students, researchers, faculty and professionals the resources they need to learn and succeed in graduate courses.
01
01
Algebra
Basic Concepts of Abstract Algebra
1
A01
01
A2354
Zhixiang WU
WU, Zhixiang
Zhixiang
WU
<p class="MsoNormal"><span lang="EN-US">Zhixiang WUis a professor at the School of Mathematical Sciences, Zhejiang University inChina. He graduated from Fudan University in China and received his Ph.D in1998. He was a visiting scholar at Wuppertal University in Germany during2003-2004 and Hamburg University in Germany during 2013-2014. He has been engagedin teaching and research in various fields of algebra, such as ring theory,homology, and Lie algebras over the years.<o:p></o:p></span></p>
01
eng
08
240
03
01
20
abstract algebra;groups;rings;modules;fields;linear representations of finite groups;Hopf algebras;Lie algebras;category theory;mathematics
10
2011
MAT002010
29
CLIL3052
01
510
05
03
00
<p>This textbook covers key topics in abstract algebra, including groups, rings, modules, and fields, as well as the linear representations of finite groups, Hopf algebras, Lie algebras, and category theory. It offers essential algebraic foundations for graduate students in mathematics and physics, and is enriched with numerous examples to facilitate understanding. It is intended for readers with a background in linear algebra.</p>
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<p>This textbook covers key topics in abstract algebra, including groups, rings, modules, and fields, as well as the linear representations of finite groups, Hopf algebras, Lie algebras, and category theory. </p>
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<p>Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III</p><p>CHAPTER 1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1</p><p>1.1 Semigroups, Monoids and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1</p><p>1.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7</p><p>1.3 The Action of a Group on a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12</p><p>1.4 The Sylow Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20</p><p>1.5 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22</p><p>1.6 Direct Products and Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30</p><p>1.7 Simple Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39</p><p>1.8 Nilpotent Groups and Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . 41</p><p>CHAPTER 2 Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47</p><p>2.1 Rings and Ring Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47</p><p>2.2 Modules, Indecomposable Modules and Free Modules . . . . . . . . . . . . . 61</p><p>2.3 Projective Modules and Injective Modules . . . . . . . . . . . . . . . . . . . . . . 74</p><p>2.4 Homological Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82</p><p>2.5 Tensor Product and Weak Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 91</p><p>2.6 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103</p><p>2.7 Noetherian Modules and UFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113</p><p>2.8 Finitely Generated Modules Over a PID . . . . . . . . . . . . . . . . . . . . . . . 124</p><p>CHAPTER 3 Fields and Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135</p><p>3.1 Extensions of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135</p><p>3.2 Splitting Fields and Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142</p><p>3.3 The Fundamental Theorem of Galois Theory . . . . . . . . . . . . . . . . . . . 151</p><p>3.4 Radical Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160</p><p>3.5 Construction with Straight-Edge and Compass . . . . . . . . . . . . . . . . . . 163</p><p>3.6 The Hilbert Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166</p><p>CHAPTER 4 Introduction to Various Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175</p><p>4.1 Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175</p><p>4.2 Coassociative Coalgebras and Hopf Algebras . . . . . . . . . . . . . . . . . . . . 188</p><p>4.3 Nonassociative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193</p><p>CHAPTER 5 Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203</p><p>5.1 Category, Limit and Colimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203</p><p>5.2 Functors and Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . . 208</p><p>5.3 Abelian Categories and Homological Groups . . . . . . . . . . . . . . . . . . . . 216</p><p>Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227</p><p>Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229</p><p>VIII Contents</p>
<p>Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III</p><p>CHAPTER 1</p><p>Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1</p><p>1.1 Semigroups, Monoids and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1</p><p>1.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7</p><p>1.3 The Action of a Group on a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12</p><p>1.4 The Sylow Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20</p><p>1.5 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22</p><p>1.6 Direct Products and Direct Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30</p><p>1.7 Simple Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39</p><p>1.8 Nilpotent Groups and Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . 41</p><p>CHAPTER 2</p><p>Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47</p><p>2.1 Rings and Ring Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47</p><p>2.2 Modules, Indecomposable Modules and Free Modules . . . . . . . . . . . . . 61</p><p>2.3 Projective Modules and Injective Modules . . . . . . . . . . . . . . . . . . . . . . 74</p><p>2.4 Homological Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82</p><p>2.5 Tensor Product and Weak Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 91</p><p>2.6 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103</p><p>2.7 Noetherian Modules and UFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113</p><p>2.8 Finitely Generated Modules Over a PID . . . . . . . . . . . . . . . . . . . . . . . 124</p><p>CHAPTER 3</p><p>Fields and Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135</p><p>3.1 Extensions of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135</p><p>3.2 Splitting Fields and Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142</p><p>3.3 The Fundamental Theorem of Galois Theory . . . . . . . . . . . . . . . . . . . 151</p><p>3.4 Radical Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160</p><p>3.5 Construction with Straight-Edge and Compass . . . . . . . . . . . . . . . . . . 163</p><p>3.6 The Hilbert Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166</p><p>CHAPTER 4</p><p>Introduction to Various Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175</p><p>4.1 Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175</p><p>4.2 Coassociative Coalgebras and Hopf Algebras . . . . . . . . . . . . . . . . . . . . 188</p><p>4.3 Nonassociative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193</p><p>CHAPTER 5</p><p>Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203</p><p>5.1 Category, Limit and Colimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203</p><p>5.2 Functors and Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . . 208</p><p>5.3 Abelian Categories and Homological Groups . . . . . . . . . . . . . . . . . . . . 216</p><p>Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227</p><p>Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229</p><p>VIII Contents</p>
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