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Elements of Abstract Algebraby Allan Clark
Synopses & ReviewsPublisher Comments:This concise, readable, collegelevel 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 JordanHolder 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. Synopsis: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. Synopsis: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. Table of Contents Foreword; Introduction
I. Set Theory 19. The notation and terminology of set theory 1016. Mappings 1719. Equivalence relations 2025. Properties of natural numbers II. Group Theory 2629. Definition of group structure 3034. Examples of group structure 3544. Subgroups and cosets 4552. Conjugacy, normal subgroups, and quotient groups 5359. The Sylow theorems 6070. Group homomorphism and isomorphism 7175. Normal and composition series 7686. The Symmetric groups III. Field Theory 8789. Definition and examples of field structure 9095. Vector spaces, bases, and dimension 9697. Extension fields 98107. Polynomials 108114. Algebraic extensions 115121. Constructions with straightedge and compass IV. Galois Theory 122126. Automorphisms 127138. Galois extensions 139149. Solvability of equations by radicals V. Ring Theory 150156. Definition and examples of ring structure 157168. Ideals 169175. Unique factorization VI. Classical Ideal Theory 176179. Fields of fractions 180187. Dedekind domains 188191. Integral extensions 192198. Algebraic integers Bibliography; Index What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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