Synopses & Reviews
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.
Helpful illustrations and exercises included throughout this lucid coverage of group theory, Galois theory and classical ideal theory stressing proof of important theorems. Includes many historical notes. Mathematical proof is emphasized. Includes 24 tables and figures. Reprint of the 1971 edition.
Lucid coverage of the major theories of abstract algebra, with helpful illustrations and exercises included throughout. Unabridged, corrected republication of the work originally published 1971. Bibliography. Index. Includes 24 tables and figures.
Table of Contents
I. Set Theory
1-9. The notation and terminology of set theory
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
108-114. Algebraic extensions
115-121. Constructions with straightedge and compass
IV. Galois Theory
127-138. Galois extensions
139-149. Solvability of equations by radicals
V. Ring Theory
150-156. Definition and examples of ring structure
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