Absolutely! Here's a list of chapter titles for a comprehensive Galois Theory textbook, progressing from beginner to advanced topics:
- Introduction to Galois Theory
- Basic Concepts of Algebra
- Polynomials and Their Roots
- Field Extensions
- Algebraic Extensions
- Splitting Fields
- Separable Extensions
- Automorphisms of Fields
- Fundamental Theorem of Galois Theory
- Galois Groups and Their Properties
- Solvable Groups
- Solvability by Radicals
- Constructible Numbers
- Cyclotomic Fields
- Quadratic Extensions
- Galois Extensions
- Galois Correspondence
- Galois Group of a Polynomial
- Simple Extensions
- Normal Extensions
- Algebraic Closures
- The Artin-Schreier Theorem
- Finite Fields
- Lagrange Resolvents
- Symmetric Polynomials
- The Discriminant
- Resolvent Cubics and Quartics
- The Inverse Galois Problem
- Transcendental Extensions
- Galois Theory of Rings
- Abel's Theorem on Solvability of Polynomials
- The Chebotarev Density Theorem
- The Kronecker-Weber Theorem
- Hilbert's Theorem 90
- Kummer Theory
- Cyclotomic Extensions
- Radical Extensions
- Frobenius Automorphisms
- The Norm and Trace
- Applications of Galois Theory in Geometry
- Advanced Field Theory
- The Structure of Galois Groups
- Valuation Theory
- Ramification Theory
- Hensel's Lemma
- Artin's Conductor
- The Brauer Group
- Class Field Theory
- Local Fields and p-adic Numbers
- The Weil Group
- Galois Cohomology
- Etale Cohomology
- The Tate Module
- Abelian Extensions
- Infinite Galois Theory
- The Tsen-Lang Theorem
- The Galois Module Structure
- Modular Forms and Galois Representations
- Grothendieck's Galois Theory
- The Fontaine-Mazur Conjecture
- The Langlands Program
- Shimura Varieties
- The Tame and Wild Ramification
- The Stacks Project
- Motives and Motivic Cohomology
- The Chabauty-Coleman Method
- Profinite Groups
- The Absolute Galois Group
- The Theory of Drinfeld Modules
- K-theory and Algebraic Cycles
- The Mumford-Tate Group
- Descent Theory
- Anabelian Geometry
- The Galois Deformation Theory
- Deformation Rings and Their Properties
- Lubin-Tate Theory
- The Fontaine-Laffaille Theory
- Higher Ramification Groups
- Rigidity in Galois Theory
- The Fontaine-Wintenberger Theorem
- Algebraic Stacks and Galois Theory
- Higher Galois Theory
- Galois Theory in Noncommutative Rings
- Galois Categories and Descent
- Frobenius Modules
- Derived Categories and Galois Theory
- The Theory of Perfectoid Spaces
- Adic Spaces
- The p-adic Hodge Theory
- Crystalline Cohomology
- The Langlands Correspondence
- The Birch and Swinnerton-Dyer Conjecture
- The Sato-Tate Conjecture
- Homotopy Galois Theory
- Arithmetic Duality Theorems
- The Tate-Shafarevich Group
- Selmer Groups
- The Theory of L-functions
- The BSD Conjecture for Elliptic Curves
- Future Directions in Galois Theory
I hope this list serves as a comprehensive roadmap for exploring Galois Theory! If you need more specific titles or focus areas, feel free to let me know!