This volume studies fields, polynomials, and algebraic extensions. It provides the structural basis for algebraic number theory, algebraic geometry, and Galois theory.
Part I. Fields and Basic Structures
Chapter 1. Fields
1.1 Definition and examples 1.2 Subfields 1.3 Field homomorphisms 1.4 Characteristic of a field 1.5 Prime fields
Chapter 2. Polynomial Rings
2.1 Polynomial definitions 2.2 Arithmetic of polynomials 2.3 Degree and leading coefficient 2.4 Division algorithm 2.5 Euclidean structure
Chapter 3. Factorization
3.1 Irreducible polynomials 3.2 Unique factorization 3.3 Eisenstein criterion 3.4 Factorization over various fields 3.5 Examples
Part II. Field Extensions
Chapter 4. Extensions of Fields
4.1 Definition of extensions 4.2 Degree of extension 4.3 Simple extensions 4.4 Algebraic vs transcendental elements 4.5 Tower law
Chapter 5. Algebraic Extensions
5.1 Minimal polynomials 5.2 Splitting fields 5.3 Algebraic closure 5.4 Construction techniques 5.5 Examples
Chapter 6. Finite Fields
6.1 Existence and uniqueness 6.2 Structure of finite fields 6.3 Polynomial factorization 6.4 Applications 6.5 Computational aspects
Part III. Galois Theory
Chapter 7. Automorphisms
7.1 Field automorphisms 7.2 Fixed fields 7.3 Groups of automorphisms 7.4 Examples 7.5 Structural properties
Chapter 8. Galois Extensions
8.1 Normal and separable extensions 8.2 Definition of Galois extensions 8.3 Fundamental theorem of Galois theory 8.4 Correspondence between subgroups and subfields 8.5 Examples
Chapter 9. Applications of Galois Theory
9.1 Solvability by radicals 9.2 Classical construction problems 9.3 Cyclotomic fields 9.4 Extensions and equations 9.5 Further applications
Part IV. Advanced Polynomial Theory
Chapter 10. Roots and Symmetry
10.1 Symmetric polynomials 10.2 Elementary symmetric functions 10.3 Relations among roots 10.4 Discriminants 10.5 Resultants
Chapter 11. Special Polynomials
11.1 Cyclotomic polynomials 11.2 Chebyshev polynomials 11.3 Orthogonal polynomials (overview) 11.4 Recursive constructions 11.5 Applications
Chapter 12. Factorization Algorithms
12.1 Polynomial factorization methods 12.2 Berlekamp algorithm 12.3 Cantor–Zassenhaus 12.4 Complexity considerations 12.5 Implementation aspects
Part V. Valuation and Local Theory
Chapter 13. Valuations
13.1 Absolute values 13.2 Non-Archimedean valuations 13.3 Completion of fields 13.4 Examples 13.5 Applications
Chapter 14. Local Fields
14.1 Definition and examples 14.2 Extensions of local fields 14.3 Ramification 14.4 Residue fields 14.5 Applications
Chapter 15. Hensel’s Lemma
15.1 Statement and proof 15.2 Lifting solutions 15.3 Applications to polynomials 15.4 p-adic factorization 15.5 Computational use
Part VI. Transcendence and Algebraic Independence
Chapter 16. Transcendental Elements
16.1 Definitions 16.2 Examples 16.3 Algebraic independence 16.4 Extensions generated by transcendental elements 16.5 Applications
Chapter 17. Transcendence Theory (Overview)
17.1 Liouville numbers 17.2 Basic transcendence results 17.3 Measures of transcendence 17.4 Methods 17.5 Examples
Chapter 18. Function Fields
18.1 Definition 18.2 Algebraic curves connection 18.3 Extensions 18.4 Divisors (overview) 18.5 Applications
Part VII. Computational and Applied Aspects
Chapter 19. Computational Field Theory
19.1 Representing fields 19.2 Arithmetic algorithms 19.3 Polynomial arithmetic 19.4 Field extensions in software 19.5 Performance considerations
Chapter 20. Applications in Coding Theory
20.1 Finite fields in coding 20.2 Reed–Solomon codes 20.3 Error detection and correction 20.4 Polynomial interpolation 20.5 Applications
Chapter 21. Applications in Cryptography
21.1 Finite field arithmetic 21.2 Elliptic curve cryptography 21.3 Pairing-based cryptography 21.4 Security assumptions 21.5 Implementation issues
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Inverse Galois problem (overview) 22.2 Infinite extensions 22.3 Field arithmetic 22.4 Model-theoretic aspects 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Classification problems 23.2 Computational complexity 23.3 Structure of extensions 23.4 Connections to geometry 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of field theory 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common polynomial identities B. Field constructions reference C. Algorithm templates D. Computational tools E. Cross-reference to other MSC branches
This volume builds the theory of fields and polynomials as a central algebraic framework. It connects structural theory with computation and applications.