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
3. Rings of Integers of Number Fields
4. Quadratic Number Fields
5. Geometric Methods
6. Localization of Dedekind Domains
7. Extensions of Dedekind Domains
8. Analytic Methods
9. Abelian Number Fields
PART-II CLASS FIELD THEORY
10. Completions of Number Fields
11. Local Fields
12. Galois Modules
13. Ray Class Groups and Dirichlet Characters
14. Artin’s Reciprocity Law
15. The Classification Theorem
16. Local Class Fields and Symbols
17. Conductor and Discriminant
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 a well-structured and comprehensive textbook designed to introduce students to "algebraic number theory" through the study of "number fields". Based on master course lecture notes at Radboud University, the book assumes only a bachelor-level understanding of mathematics and basic "Galois theory". It begins with foundational concepts, including ring and field theory, ideals, and the arithmetic of algebraic integers, gradually building the reader’s understanding of the structures and properties that underpin number fields. The text emphasizes clarity and logical progression, making complex ideas accessible to both classroom learners and self-studying students.
The book progresses to more advanced topics, such as "class field theory", Dirichlet characters, and explicit constructions of abelian extensions. Keune provides full proofs of important theorems and illustrates concepts with examples and exercises. Algorithmic methods for computing class groups, units, and prime decomposition in quadratic number fields are also included, helping readers see the practical side of theoretical concepts.
Designed to be self-contained, the textbook integrates classical approaches with modern mathematical language. It balances theory with computation, fostering both deep understanding and problem-solving skills. Freely available under a Creative Commons license, "Number Fields" serves as an excellent resource for anyone studying algebraic number theory, whether in a formal classroom setting or independently.
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 a Dutch mathematician and professor at "Radboud University Nijmegen", recognized for his expertise in "algebraic number theory" and decades of teaching advanced mathematics. He is known for making abstract concepts clear and accessible for graduate students and self-learners alike. His text, "Number Fields", is a self-contained introduction to algebraic number theory covering "number fields", ideals, Dirichlet characters, and "class field theory". The text is ideal for students seeking a solid foundation in "algebraic number theory" and its classical applications.
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