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Number Theory in Function Fields

  • Textbook
  • © 2002

Overview

Authors:
  1. Michael Rosen

    1. Department of Mathematics, Brown University, Providence, USA

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Part of the book series: Graduate Texts in Mathematics (GTM, volume 210)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xii
    Download chapter PDF

  2. Polynomials over Finite Fields

    • Michael Rosen
    Pages 1-9

  3. Primes, Arithmetic Functions, and the Zeta Function

    • Michael Rosen
    Pages 11-21

  4. The Reciprocity Law

    • Michael Rosen
    Pages 23-31

  5. Dirichlet L-Series and Primes in an Arithmetic Progression

    • Michael Rosen
    Pages 33-43

  6. Algebraic Function Fields and Global Function Fields

    • Michael Rosen
    Pages 45-61

  7. Weil Differentials and the Canonical Class

    • Michael Rosen
    Pages 63-76

  8. Extensions of Function Fields, Riemann-Hurwitz, and the ABC Theorem

    • Michael Rosen
    Pages 77-99

  9. Constant Field Extensions

    • Michael Rosen
    Pages 101-113

  10. Galois Extensions — Hecke and Artin L-Series

    • Michael Rosen
    Pages 115-147

  11. Artin’s Primitive Root Conjecture

    • Michael Rosen
    Pages 149-167

  12. The Behavior of the Class Group in Constant Field Extensions

    • Michael Rosen
    Pages 169-191

  13. Cyclotomic Function Fields

    • Michael Rosen
    Pages 193-217

  14. Drinfeld Modules: An Introduction

    • Michael Rosen
    Pages 219-239

  15. S-Units, S-Class Group, and the Corresponding L-Functions

    • Michael Rosen
    Pages 241-256

  16. The Brumer-Stark Conjecture

    • Michael Rosen
    Pages 257-281

  17. The Class Number Formulas in Quadratic and Cyclotomic Function Fields

    • Michael Rosen
    Pages 283-303

  18. Average Value Theorems in Function Fields

    • Michael Rosen
    Pages 305-327

  19. Back Matter

    Pages 329-361
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Keywords

About this book

Elementary number theory is concerned with the arithmetic properties of the ring of integers, Z, and its field of fractions, the rational numbers, Q. Early on in the development of the subject it was noticed that Z has many properties in common with A = IF[T], the ring of polynomials over a finite field. Both rings are principal ideal domains, both have the property that the residue class ring of any non-zero ideal is finite, both rings have infinitely many prime elements, and both rings have finitely many units. Thus, one is led to suspect that many results which hold for Z have analogues of the ring A. This is indeed the case. The first four chapters of this book are devoted to illustrating this by presenting, for example, analogues of the little theorems of Fermat and Euler, Wilson's theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlet's theorem on primes in an arithmetic progression. All these results have been known for a long time, but it is hard to locate any exposition of them outside of the original papers. Algebraic number theory arises from elementary number theory by con­ sidering finite algebraic extensions K of Q, which are called algebraic num­ ber fields, and investigating properties of the ring of algebraic integers OK C K, defined as the integral closure of Z in K.

Reviews

From the reviews:

MATHEMATICAL REVIEWS

"Both in the large (choice and arrangement of the material) and in the details (accuracy and completeness of proofs, quality of explanations and motivating remarks), the author did a marvelous job. His parallel treatment of topics…for both number and function fields demonstrates the strong interaction between the respective arithmetics, and allows for motivation on either side."

Bulletin of the AMS

"… Which brings us to the book by Michael Rosen. In it, one has an excellent (and, to the author's knowledge, unique) introduction to the global theory of function fields covering both the classical theory of Artin, Hasse, Weil and presenting an introduction to Drinfeld modules (in particular, the Carlitz module and its exponential). So the reader will find the basic material on function fields and their history (i.e., Weil differentials, the Riemann-Roch Theorem etc.) leading up to Bombieri's proof of the Riemann hypothesis first established by Weil. In addition one finds chapters on Artin's primitive root Conjecture for function fields, Brumer-Stark theory, the ABC Conjecture, results on class numbers and so on. Each chapter contains a list of illuminating exercises. Rosen's book is perfect for graduate students, as well as other mathematicians, fascinated by the amazing similarities between number fields and function fields."

David Goss (Ohio State University)

Authors and Affiliations

  • Department of Mathematics, Brown University, Providence, USA

    Michael Rosen

Bibliographic Information

  • Book Title: Number Theory in Function Fields

  • Authors: Michael Rosen

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4757-6046-0

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 2002

  • Hardcover ISBN: 978-0-387-95335-9Published: 08 January 2002

  • Softcover ISBN: 978-1-4419-2954-9Published: 03 December 2010

  • eBook ISBN: 978-1-4757-6046-0Published: 18 April 2013

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: XI, 358

  • Topics: Number Theory, Algebraic Geometry, Field Theory and Polynomials

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