Number Fields by Frans Keune
Number Fields - Table of Contents
PART I: BASIC ALGEBRAIC NUMBER THEORY
- 1. Integers in a Number Field
- 2. Dedekind Domains and Ideal Factorization
- 3. Rings of Integers of Number Fields
- 4. Quadratic Number Fields
- 5. Geometric Methods and Geometry of Numbers
- 6. Localization of Dedekind Domains
- 7. Extensions of Dedekind Domains
- 8. Analytic Methods and Zeta Functions
- 9. Abelian Number Fields
PART II-A: LOCAL FIELDS & GROUPS
- 10. Completions of Number Fields
- 11. Local Fields and Valuation Theory
- 12. Galois Modules and Cohomology
- 13. Ray Class Groups and Dirichlet Characters
- 14. Artin’s Reciprocity Law
PART II-B: CLASS FIELD THEORY
- 15. The Classification Theorem of Class Fields
- 16. Local Class Fields and Symbols
- 17. Conductor and Discriminant Relations
- 18. Zeta Function Relations
- 19. Infinite Extensions of Number Fields
- 20. Idèlic Class Field Theory
What You Will Learn in Number Fields
Number Fields by Frans Keune is an authoritative, mathematically rigorous advanced textbook published by Springer that provides graduate students with a complete structural treatment of modern algebraic number theory. This comprehensive volume guides readers through rings of integers, Dedekind domain localizations, Minkowski geometry of numbers, Dirichlet L-functions, local fields, ray class groups, Galois cohomology, Artin's reciprocity law, and idèle class groups with exceptional precision and structural clarity.
Designed for graduate mathematics students, researchers in arithmetic geometry, and self-study algebraists, this advanced text connects classical algebraic number theory directly to modern class field theory. Keune details step-by-step mathematical proofs for prime ideal factorizations, class number finiteness, local and global reciprocity, conductor-discriminant formulas, and infinite Galois extensions. Whether you are studying class field classification theorems, exploring p-adic field completions, or researching modular arithmetic structures, this book offers an indispensable roadmap.
Recognized globally for its elegant structural exposition and unified idelic approach, it stands as one of the best advanced books on number fields and class field theory. By balancing global geometry with local field completions, it serves as an essential tool for mastering algebraic number theory and arithmetic structures with confidence.
Book Details & Specifications
Title:
Number Fields by Frans Keune
Publisher:
Radboud University Press
Year:
2023
Pages:
587
Type:
PDF
Language:
English
ISBN-10 #:
9493296032
ISBN-13 #:
978-9493296039
License:
CC BY-NC-ND 4.0
Amazon:
Amazon
About the Author: Frans Keune
The author Frans Keune
is Emeritus Associate Professor of Mathematics at Radboud University Nijmegen, Netherlands, internationally recognized for his research contributions to algebraic K-theory, class field theory, and number theory. He has written multiple graduate-level texts celebrated for their structural rigor, clarity, and pedagogical depth.
Through Number Fields, Keune delivers a masterful, self-contained exposition of class field theory and modern arithmetic. His lucid proof structures, unified idelic formulation, and systematic treatment of Galois modules make this volume a classic reference for algebraic number theory trusted by researchers and graduate students worldwide.
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