Here’s a list of 100 chapter titles for a book or course on Field Theory, progressing from beginner to advanced topics, with a focus on mathematical concepts:
- Introduction to Field Theory: Concepts and Applications
- What is a Field? Basic Definitions and Examples
- Algebraic Structures: Sets, Operations, and Field Properties
- The Axioms of a Field
- Examples of Fields: Rational Numbers, Real Numbers, and Complex Numbers
- Field Extensions: The Basic Idea
- Subfields: Definition and Examples
- Field Operations: Addition, Multiplication, and Inverses
- The Field of Rational Numbers: A First Example
- The Field of Real Numbers and Its Properties
- The Field of Complex Numbers and Their Role in Mathematics
- Field Homomorphisms and Automorphisms
- Finite Fields: Definition and Examples
- Understanding Field Characteristics
- Field Equations and Polynomial Roots
- The Field of Polynomials: Basics and Operations
- Introduction to Algebraic Numbers
- Field Extensions and Minimal Polynomials
- Degree of a Field Extension
- Roots of Polynomials and Fields of Algebraic Numbers
- Field of Rational Functions: Definition and Examples
- Constructing Finite Fields
- Applications of Field Theory in Number Theory
- Field Extensions and Splitting Fields
- Transcendental Extensions and Their Properties
- The Concept of Algebraic Closure
- Field Extensions in Abstract Algebra
- Field Homomorphisms: The Role in Field Theory
- Introduction to Galois Theory
- Basic Examples of Field Extensions: Q ⟶ R, Q ⟶ C
- Polynomial Rings and Fields
- Irreducibility Criteria for Polynomials
- Constructing Field Extensions Using Roots of Polynomials
- Finite Field Theory and Applications
- The Fundamental Theorem of Algebra
- Galois Groups and Field Extensions
- Normal and Separable Extensions
- Simple Field Extensions and Their Properties
- The Tower Law for Field Extensions
- Splitting Fields and Their Applications
- The Galois Correspondence Theorem
- The Galois Group and its Action on Roots
- Solvability by Radicals and Galois Theory
- Algebraic Closure and Its Importance in Field Theory
- Transcendental Numbers and Extensions
- Minimal Polynomials and their Importance in Field Theory
- Degree of Field Extensions and Their Computation
- Applications of Galois Theory in Solving Polynomials
- Cubic and Quartic Equations: The Role of Field Extensions
- Field Extensions of Finite Fields: Structure and Applications
- Field Automorphisms and Their Structure
- Structure of Galois Groups: Cyclic and Non-Cyclic Groups
- Applications of Field Extensions in Cryptography
- Basic Applications of Field Theory in Geometry
- Field Extensions in the Context of Algebraic Geometry
- Fields in Commutative Algebra
- Structure of Field Extensions and Their Subfields
- Classification of Field Extensions: Algebraic vs. Transcendental
- Complexity of Field Extensions: Degree and Dimension
- Solvable Groups and Their Relation to Field Theory
- The Galois Group of Polynomials
- Construction of Finite Extensions of Finite Fields
- Algebraic Number Fields and Their Extensions
- Simple and Composite Extensions in Galois Theory
- Root-Finding Algorithms Using Field Extensions
- Symmetric Functions and Galois Theory
- Field Extensions in Cryptography and Error-Correcting Codes
- Degree of Extensions in Algebraic Numbers
- Classical Problems in Galois Theory
- Rational Functions and Field Extensions
- Finite Field Theory and Its Applications in Coding Theory
- Understanding the Concept of Algebraic Closure in Detail
- Primitive Elements and Their Role in Field Extensions
- Radical Extensions and Their Relationship to Galois Groups
- Applications of Galois Theory in Solving Polynomial Equations
- Advanced Topics in Galois Theory
- Advanced Field Extensions: Infinite vs. Finite Extensions
- Transcendence Basis of a Field Extension
- The Structure of Galois Groups: Abelian and Non-Abelian
- Advanced Algebraic Geometry and Field Theory
- Field Theory and Its Relationship to Ring Theory
- Fields of Algebraic Numbers and Their Applications
- Applications of Galois Theory in Class Field Theory
- The Structure of the Absolute Galois Group
- Field Extensions and Their Interaction with Algebraic Structures
- Advanced Computational Methods for Field Extensions
- The Theory of Invariant Theory and Its Connection to Fields
- The Brauer Group and Field Theory
- Exploring Fields in High-Dimensional Geometry
- Field Extensions and Their Application to Modular Forms
- Algebraic Structures in the Context of Field Extensions
- Advanced Techniques in Solving Polynomial Equations Using Galois Theory
- The Role of Group Theory in Field Extensions
- Field Theory and Its Role in the Study of Algebraic Groups
- The Connection Between Field Theory and Algebraic Topology
- Advanced Applications of Field Extensions in Number Theory
- Infinite Galois Theory: Advanced Topics and Techniques
- Explicit Computation of Galois Groups for Large Fields
- Advanced Methods in Class Field Theory and Galois Theory
- Field Theory and Its Applications to Algebraic K-Theory
These chapters provide a well-rounded progression through Field Theory, from fundamental definitions and examples to more advanced applications in areas such as Galois Theory, cryptography, algebraic geometry, and class field theory.