Here are 100 chapter titles for a book on Computational Algebra, progressing from beginner to advanced:
I. Foundations (20 Chapters)
- Introduction to Abstract Algebra
- Sets, Relations, and Functions
- Basic Number Theory: Integers and Divisibility
- Modular Arithmetic and Congruences
- Groups: Definitions and Examples
- Subgroups and Cyclic Groups
- Group Homomorphisms and Isomorphisms
- Permutation Groups and Cayley's Theorem
- Rings: Definitions and Examples
- Integral Domains and Fields
- Polynomials: Basic Operations
- Polynomial Rings and Factorization
- Vector Spaces and Linear Algebra
- Matrices and Matrix Operations
- Linear Transformations and Eigenvalues
- Introduction to Algorithms and Complexity
- Big O Notation and Algorithm Analysis
- Data Structures for Algebraic Computation
- Introduction to Computer Algebra Systems (CAS)
- Basic Programming for Algebraic Tasks
II. Core Computational Algebra (30 Chapters)
- Euclidean Algorithm and GCD Computation
- Extended Euclidean Algorithm and Linear Diophantine Equations
- Modular Arithmetic Algorithms
- Fast Integer Arithmetic
- Polynomial Arithmetic: Addition, Multiplication, Division
- Polynomial GCD and Factorization Algorithms
- Solving Linear Systems over Fields
- Gaussian Elimination and LU Decomposition
- Matrix Operations and Algorithms
- Computing Determinants and Inverses
- Eigenvalue and Eigenvector Computation
- Introduction to Gröbner Bases
- Buchberger's Algorithm
- Applications of Gröbner Bases: Ideal Membership
- Applications of Gröbner Bases: Solving Polynomial Systems
- Applications of Gröbner Bases: Elimination Theory
- Computational Aspects of Field Extensions
- Finite Fields and Their Arithmetic
- Polynomial Factorization over Finite Fields
- Introduction to Algebraic Geometry
- Affine Varieties and Ideals
- Computational Aspects of Algebraic Varieties
- Solving Systems of Algebraic Equations
- Symbolic Computation and Manipulation
- Representation of Algebraic Expressions
- Simplification of Algebraic Expressions
- Differentiation and Integration of Symbolic Expressions
- Introduction to Computational Group Theory
- Representations of Groups
- Computational Aspects of Group Actions
III. Advanced Topics and Applications (30 Chapters)
- Advanced Gröbner Basis Techniques
- Strategies for Improving Buchberger's Algorithm
- Applications of Gröbner Bases: Robotics and Coding Theory
- Computational Algebraic Geometry: Advanced Topics
- Projective Geometry and Projective Varieties
- Computational Aspects of Projective Varieties
- Toric Varieties and Their Computation
- Computational Invariant Theory
- Representation Theory: Computational Aspects
- Character Tables and Their Computation
- Computational Aspects of Lie Algebras
- Lie Groups and Their Representations
- Computational Homological Algebra
- Simplicial Homology and its Computation
- Computational Topology
- Knot Theory and its Computational Aspects
- Computational Number Theory: Advanced Topics
- Elliptic Curves and Cryptography
- Integer Factorization Algorithms
- Primality Testing Algorithms
- Computational Algebraic Number Theory
- Algebraic Function Fields and Their Computation
- Applications of Computational Algebra in Cryptography
- Applications of Computational Algebra in Coding Theory
- Applications of Computational Algebra in Robotics
- Applications of Computational Algebra in Physics
- Applications of Computational Algebra in Biology
- Parallel Algorithms for Algebraic Computation
- Distributed Algorithms for Algebraic Computation
- High-Performance Computing for Algebraic Tasks
IV. Further Explorations and Specialized Topics (20 Chapters)
- Computational Complexity of Algebraic Algorithms
- Lower Bounds for Algebraic Computation
- Probabilistic Algorithms in Algebra
- Quantum Algorithms for Algebraic Problems
- Symbolic-Numeric Computation
- Hybrid Methods in Computational Algebra
- Software Packages for Computational Algebra
- SageMath: A Comprehensive CAS
- Magma: A Specialized CAS for Algebra
- GAP: A System for Computational Group Theory
- Singular: A System for Polynomial Computations
- Developing Custom Algorithms for Algebraic Problems
- Case Studies in Computational Algebra
- Open Problems in Computational Algebra
- Future Directions in Computational Algebra
- Connections to Other Areas of Mathematics
- Computational Algebra and Machine Learning
- Computational Algebra and Data Science
- Historical Development of Computational Algebra
- Appendix: Foundational Material and References