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.
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
