Elements of Abstract Algebra

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Overview

This concise, readable, college-level text treats basic abstract algebra in remarkable depth and detail. An antidote to the usual surveys of structure, the book presents group theory, Galois theory, and classical ideal theory in a framework emphasizing proof of important theorems.
Chapter I (Set Theory) covers the basics of sets. Chapter II (Group Theory) is a rigorous introduction to groups. It contains all the results needed for Galois theory as well as the Sylow theorems, the Jordan-Holder theorem, and a complete treatment of the simplicity of alternating groups. Chapter III (Field Theory) reviews linear algebra and introduces fields as a prelude to Galois theory. In addition there is a full discussion of the constructibility of regular polygons. Chapter IV (Galois Theory) gives a thorough treatment of this classical topic, including a detailed presentation of the solvability of equations in radicals that actually includes solutions of equations of degree 3 and 4 ― a feature omitted from all texts of the last 40 years. Chapter V (Ring Theory) contains basic information about rings and unique factorization to set the stage for classical ideal theory. Chapter VI (Classical Ideal Theory) ends with an elementary proof of the Fundamental Theorem of Algebraic Number Theory for the special case of Galois extensions of the rational field, a result which brings together all the major themes of the book.
The writing is clear and careful throughout, and includes many historical notes. Mathematical proof is emphasized. The text comprises 198 articles ranging in length from a paragraph to a page or two, pitched at a level that encourages careful reading. Most articles are accompanied by exercises, varying in level from the simple to the difficult.

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Product Details

  • ISBN-13: 9780486647258
  • Publisher: Dover Publications
  • Publication date: 10/1/1984
  • Series: Dover Books on Mathematics Series
  • Pages: 224
  • Sales rank: 538,185
  • Product dimensions: 5.30 (w) x 8.40 (h) x 0.50 (d)

Table of Contents

  Foreword; Introduction
I. Set Theory
  1-9. The notation and terminology of set theory
  10-16. Mappings
  17-19. Equivalence relations
  20-25. Properties of natural numbers
II. Group Theory
  26-29. Definition of group structure
  30-34. Examples of group structure
  35-44. Subgroups and cosets
  45-52. Conjugacy, normal subgroups, and quotient groups
  53-59. The Sylow theorems
  60-70. Group homomorphism and isomorphism
  71-75. Normal and composition series
  76-86. The Symmetric groups
III. Field Theory
  87-89. Definition and examples of field structure
  90-95. Vector spaces, bases, and dimension
  96-97. Extension fields
  98-107. Polynomials
  108-114. Algebraic extensions
  115-121. Constructions with straightedge and compass
IV. Galois Theory
  122-126. Automorphisms
  127-138. Galois extensions
  139-149. Solvability of equations by radicals
V. Ring Theory
  150-156. Definition and examples of ring structure
  157-168. Ideals
  169-175. Unique factorization
VI. Classical Ideal Theory
  176-179. Fields of fractions
  180-187. Dedekind domains
  188-191. Integral extensions
  192-198. Algebraic integers
  Bibliography; Index
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