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2018 | Book

A Special Case of Fermat’s Conjecture Read chapter Read first chapter

Number Fields

Author: Daniel A. Marcus

Publisher: Springer International Publishing

Book Series : Universitext

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Requiring no more than a basic knowledge of abstract algebra, this textbook presents the basics of algebraic number theory in a straightforward, "down-to-earth" manner. It thus avoids local methods, for example, and presents proofs in a way that highlights key arguments. There are several hundred exercises, providing a wealth of both computational and theoretical practice, as well as appendices summarizing the necessary background in algebra.

Now in a newly typeset edition including a foreword by Barry Mazur, this highly regarded textbook will continue to provide lecturers and their students with an invaluable resource and a compelling gateway to a beautiful subject.

From the reviews:

“A thoroughly delightful introduction to algebraic number theory” – Ezra Brown in the Mathematical Reviews

“An excellent basis for an introductory graduate course in algebraic number theory” – Harold Edwards in the Bulletin of the American Mathematical Society

Table of Contents

Frontmatter
Chapter 1. A Special Case of Fermat’s Conjecture
Abstract
Fermat’s last theorem is used to motivate the introduction of certain number fields.
Daniel A. Marcus
Chapter 2. Number Fields and Number Rings
Abstract
Basic properties of number fields are studied, including special results for cyclotomic fields.
Daniel A. Marcus
Chapter 3. Prime Decomposition in Number Rings
Abstract
Ideals in number rings are studied: prime decomposition, ramification, residual degrees.
Daniel A. Marcus
Chapter 4. Galois Theory Applied to Prime Decomposition
Abstract
Galois theory is applied to the general problem of determining how a prime ideal of a number rings splits in an extension field.
Daniel A. Marcus
Chapter 5. The Ideal Class Group and the Unit Group
Abstract
The geometry of numbers is used to prove results about the ideal class group of a number ring, in particular its finiteness, and determine the structure of the group of units of a number ring.
Daniel A. Marcus
Chapter 6. The Distribution of Ideals in a Number Ring
Abstract
Ideals are shown to be distributed approximately equally among the ideal classes.
Daniel A. Marcus
Chapter 7. The Dedekind Zeta Function and the Class Number Formula
Abstract
The results of chapter 6 are used to define and establish properties of the number fields and their zeta functions, such as the Class Number Formula.
Daniel A. Marcus
Chapter 8. The Distribution of Primes and an Introduction to Class Field Theory
Abstract
A general result on abstract zeta functions is established and applied to deduce results on the distribution of prime ideals. Facts from Class Field Theory are introduced without proof for this purpose.
Daniel A. Marcus
Backmatter
Metadata
Title
Number Fields
Author
Daniel A. Marcus
Copyright Year
2018
Electronic ISBN
978-3-319-90233-3
Print ISBN
978-3-319-90232-6
DOI
https://doi.org/10.1007/978-3-319-90233-3

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