Algebraic Number Theory
Authors
Description
Number theory is, in principle, a theory of rational and integer numbers and is largely linked to the problem of solving Diophantine equations, i.e. finding integer solutions to algebraic equations.
The prerequisites for reading this book have been reduced to the most basic knowledge of algebra. The basic notions about bodies, rings and modules that will be used are found without demonstration (but sometimes with references) in §0. Additional results, including those usually covered in a Master’s course in Algebra, will be demonstrated in the paragraphs where they are used.
It is hoped that this book will be useful for spreading the word about algebraic numbers in Brazil and stimulating deeper studies in Number Theory, which is often called the “Queen of Mathematics”. It can also serve as a gentle and well-motivated introduction to some topics in Commutative Algebra, as it introduces the abstract notions of this theory to be immediately applied to the concrete case of algebraic numbers.
Target audience
Higher education
Name: Algebraic Number Theory
Author(s): e Otto Endler
Pages: 199
Publication: IMPA, 2014
ISBN: 978-85-244-0026-1
Edition: 2
Prologue
CHAPTER I – ALGEBRAIC NUMBER BODIES
0. Basic notions about bodies, rings and modules
1. The IL ring of algebraic integers
2. Quadratic bodies
3. Cyclotomic bodies
4. Discriminant
5. Integral bases
CHAPTER II – NOETHERIAN RINGS AND DEDEKIND DOMAINS
6. The Chinese remainder theorem
7. Noetherian rings and Noetherian modules
8. Dedekind domains
CHAPTER III – CLASSES OF IDEALS
9. Norm of ideals
10. Finitude of the number of classes
CHAPTER IV – EXTENSIONS OF DEDEKIND DOMAINS
11. Rings of fractions of a domain
12. Decomposition of prime ideals
13. A theorem of Kummer. Ramification
CHAPTER V – DECOMPOSITION INTO CYCLOTOMIC AND QUADRATIC BODIES
14. Decomposition into cyclotomic bodies
15. Decomposition into quadratic bodies
16. Quadratic reciprocity
CHAPTER VI – THE GEOMETRIC METHOD
17. Networks in Rn
18. Geometric representations of algebraic numbers
19. Invertibles in quadratic bodies
CHAPTER VII – GALOISIAN EXTENSIONS
20. Decomposition groups and bodies
21. Inertia and branching groups and bodies
Epilogue
References
Index of notations
Alphabetical index