Workbook in Higher Algebra

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• Contents
• Acknowledgement
• 1.1 Review of Important Basics
• 1.2 The Concept of a Group Action
• 1.3 Sylow’s Theorem
• 1.4 Examples: The Linear Groups
• 1.5 Automorphism Groups and the Semi-Direct Product
• 1.6 The Symmetric and Alternating Groups
• 1.7 The Commutator Subgroup and Iterated Constructions
• 1.8 Free Groups; Generators and Relations
• 2.1 Basics
• 2.2 Splitting Fields and Algebraic Closure
• 2.3 Galois Extensions, Galois Groups and the Fundamental Theorem of Galois Theory
• 2.4 Separability and the Galois Criterion
• 2.5 Brief Interlude: the Krull Topology
• 2.6 The Fundamental Theorem of Algebra
• 2.7 The Galois Group of a Polynomial
• 2.8 The Cyclotomic Polynomials
• 2.9 Solvability by Radicals
• 2.10 The Primitive Element Theorem
• 3.1 Basics
• 3.2 Unique Factorization Domains
• 3.3 Noetherian Rings and Principal Ideal Domains
• 3.4 Principal Ideal Domains and Euclidean Domains
• 4.1 A Few Remarks About Module Theory
• 4.2 Algebraic Integer Domains
• 4.3 OE is a Dedekind Domain
• 4.4 Factorization Theory in Dedekind Domains and the Fundamental Theorem of Algebraic Number Theory
• 4.5 The Ideal Class Group of a Dedekind Domain
• 4.6 A Characterization of Dedekind Domains
• 5.1 The Basic Homomorphism Theorems
• 5.2 Direct Products and Sums of Modules; Free Modules
• 5.3 Modules over a Principal Ideal Domain
• 5.4 Calculation of Invariant Factors
• 5.5 Application to a Single Linear Transformation
• 5.6 Chain Conditions and Series of Modules
• 5.7 The Krull-Schmidt Theorem
• 5.8 Injective and Projective Modules
• 5.9 Semisimple Modules
• 5.10 Example: Group Algebras
• 6.1 The Jacobson Radical and Semisimple Artinian Rings
• 7.1 Tensor Product as an Abelian Group
• 7.2 Tensor Product as a Left S-Module
• 7.3 Tensor Product as an Algebra
• 7.4 Tensor, Symmetric and Exterior Algebra of a Vector Space
• 7.5 The Adjointness Relationship
• Appendix A Zorn’s Lemma and some Applications

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