A Field Guide to Algebra

  • Antoine Chambert-Loir

Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages I-X
  2. Pages 1-30
  3. Pages 31-53
  4. Pages 55-81
  5. Pages 83-105
  6. Pages 107-149
  7. Back Matter
    Pages 181-195

About this book

Introduction

This unique textbook focuses on the structure of fields and is intended for a second course in abstract algebra. Besides providing proofs of the transcendance of pi and e, the book includes material on differential Galois groups and a proof of Hilbert's irreducibility theorem. The reader will hear about equations, both polynomial and differential, and about the algebraic structure of their solutions. In explaining these concepts, the author also provides comments on their historical development and leads the reader along many interesting paths.

In addition, there are theorems from analysis: as stated before, the transcendence of the numbers pi and e, the fact that the complex numbers form an algebraically closed field, and also Puiseux's theorem that shows how one can parametrize the roots of polynomial equations, the coefficients of which are allowed to vary. There are exercises at the end of each chapter, varying in degree from easy to difficult. To make the book more lively, the author has incorporated pictures from the history of mathematics, including scans of mathematical stamps and pictures of mathematicians.

Antoine Chambert-Loir taught this book when he was Professor at École polytechnique, Palaiseau, France. He is now Professor at Université de Rennes 1.

Keywords

Abstract algebra Algebraic structure Group theory Irreducibility algebra

Authors and affiliations

  • Antoine Chambert-Loir
    • 1
  1. 1.IRMARUniversité de Rennes 1Rennes CedexFrance

Bibliographic information

  • DOI https://doi.org/10.1007/b138364
  • Copyright Information Springer Science+Business Media, Inc. 2005
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-387-21428-3
  • Online ISBN 978-0-387-26955-9
  • Series Print ISSN 0172-6056
  • About this book