1CHAPTER 1 Fundamental Algebraic Structures
58CHAPTER 9 Homological Algebra
21.1. Sets and Set Operations
599.1. Exact Sequences and Derived Functors
31.2. Relations and Functions
609.2. Projective and Injective Modules
41.3. Algebraic Systems: Groups, Rings, and Fields
619.3. Ext and Tor Functors
51.4. Homomorphisms and Isomorphisms
629.4. Spectral Sequences and their Applications
61.5. Substructures and Quotient Structures
639.5. Cohomology of Groups and Algebras
71.6. Examples and Applications of Algebraic Structures
649.6. Connections to Algebraic Geometry and Topology
8CHAPTER 2 Groups
65CHAPTER 10 Algebraic Number Theory
92.1. Definition and Properties of Groups
6610.1. Algebraic Integers and Ideals
102.2. Abelian Groups and Cyclic Groups
6710.2. Dedekind Domains and Factorization
112.3. Cosets, Lagrange’s Theorem, and Quotient Groups
6810.3. Class Groups and the Reciprocity Law
122.4. Group Homomorphisms and Isomorphisms
6910.4. Valuations and Local-Global Principles
132.5. Normal Subgroups and Factor Groups
7010.5. Diophantine Equations and Fermat’s Last Theorem
142.6. Permutation Groups and Group Actions
7110.6. Applications in Cryptography and Computer Science
152.7. Applications of Group Theory
72CHAPTER 11 Algebraic Geometry
16CHAPTER 3 Rings and Modules
7311.1. Affine and Projective Varieties
173.1. Definition and Properties of Rings
7411.2. Morphisms and Rational Maps
183.2. Ideals and Homomorphisms
7511.3. Sheaves and the Zariski Topology
193.3. Polynomial Rings and Unique Factorization Domains
7611.4. Divisors and the Riemann-Roch Theorem
203.4. Principal Ideal Domains and Euclidean Domains
7711.5. Algebraic Curves and Surfaces
213.5. Modules and Module Homomorphisms
7811.6. Applications in Physics and Engineering
223.6. Structure Theorems for Finitely Generated Modules
79CHAPTER 12 Lie Algebras and Lie Groups
233.7. Applications of Ring and Module Theory
8012.1. Definition and Properties of Lie Algebras
24CHAPTER 4 Fields and Galois Theory
8112.2. Representations of Lie Algebras
254.1. Definition and Properties of Fields
8212.3. Universal Enveloping Algebras and the Poincaré-Birkhoff-Witt Theorem
264.2. Finite Fields and Algebraic Extensions
8312.4. Classification of Semisimple Lie Algebras
274.3. Splitting Fields and Normal Extensions
8412.5. Lie Groups and their Relationship to Lie Algebras
284.4. Galois Groups and the Fundamental Theorem of Galois Theory
8512.6. Applications in Theoretical Physics and Geometry
294.5. Solvability by Radicals and the Insolvability of the Quintic
86CHAPTER 13 Noncommutative Algebra
304.6. Applications of Galois Theory
8713.1. Associative Algebras and their Representations
31CHAPTER 5 Linear Algebra
8813.2. Jacobson Radical and Semisimple Algebras
325.1. Vector Spaces and Subspaces
8913.3. Quivers and Representation Theory
335.2. Linear Transformations and Matrix Representations
9013.4. Geometric Invariant Theory and Moduli Spaces
345.3. Eigenvalues, Eigenvectors, and Diagonalization
9113.5. Noncommutative Algebraic Geometry
355.4. Inner Product Spaces and Orthogonality
9213.6. Applications in Quantum Mechanics and Noncommutative Geometry
365.5. Determinants and the Rank-Nullity Theorem
93CHAPTER 14 Algebraic Combinatorics
375.6. Canonical Forms and the Jordan Canonical Form
9414.1. Posets, Lattices, and Möbius Functions
38CHAPTER 6 Multilinear Algebra
9514.2. Symmetric Functions and Schur Polynomials
396.1. Tensor Products of Vector Spaces
9614.3. Matroids and Combinatorial Optimization
406.2. Exterior Algebra and Alternating Multilinear Forms
9714.4. Algebraic Enumeration and Generating Functions
416.3. Determinants and the Exterior Algebra
9814.5. Connections to Representation Theory and Algebraic Geometry
426.4. Applications of Multilinear Algebra in Geometry and Physics
9914.6. Applications in Theoretical Computer Science and Coding Theory
436.5. Tensor Fields and Differential Geometry
100CHAPTER 15 Category Theory
44CHAPTER 7 Advanced Group Theory
10115.1. Categories, Functors, and Natural Transformations
457.1. Sylow Theorems and Applications
10215.2. Universal Constructions and Adjoint Functors
467.2. Solvable and Nilpotent Groups
10315.3. Limits and Colimits
477.3. Simple Groups and the Classification Theorem
10415.4. Abelian Categories and the Derived Category
487.4. Representations of Finite Groups
10515.5. Applications in Algebra, Topology, and Logic
497.5. Character Theory and the Artin-Wedderburn Theorem
10615.6. Categorical Approach to Algebraic Structures
507.6. Applications of Advanced Group Theory
107CHAPTER 16 Algebraic Topology
51CHAPTER 8 Commutative Algebra
10816.2. Homology and Cohomology Theories
528.1. Noetherian Rings and Modules
10916.3. Homotopy Theory and the Hurewicz Theorem
538.2. Primary Decomposition and Integral Dependence
11016.4. Spectral Sequences and Computational Techniques
548.3. Localization and Completion
11116.5. Applications in Geometry, Physics, and Computer Science
558.4. Dimension Theory and the Hilbert Basis Theorem
11216.6. Connections to Algebraic Geometry and Commutative Algebra
568.5. Affine Algebraic Geometry and Spec(R)
113Index
578.6. Applications in Algebraic Geometry and Algebraic Number Theory
