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ویرایش:
نویسندگان: Gouvea F.Q.
سری:
ISBN (شابک) : 9780883853559
ناشر: MAA
سال نشر: 2012
تعداد صفحات: 328
زبان: English
فرمت فایل : DJVU (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود)
حجم فایل: 2 مگابایت
در صورت تبدیل فایل کتاب A guide to groups, rings, and fields به فرمت های PDF، EPUB، AZW3، MOBI و یا DJVU می توانید به پشتیبان اطلاع دهید تا فایل مورد نظر را تبدیل نمایند.
توجه داشته باشید کتاب راهنمای گروه ها، حلقه ها و زمینه ها نسخه زبان اصلی می باشد و کتاب ترجمه شده به فارسی نمی باشد. وبسایت اینترنشنال لایبرری ارائه دهنده کتاب های زبان اصلی می باشد و هیچ گونه کتاب ترجمه شده یا نوشته شده به فارسی را ارائه نمی دهد.
ساختارهای جبری در ریاضیات همه جا حاضر شده اند، تقریباً همه ریاضیدانان در طول دوره تحقیقات خود با گروه ها، حلقه ها، میدان ها یا اشیاء عجیب و غریب مرتبط بیشتری روبرو می شوند. این کتاب مروری بر برخی از مهمترین ساختارهای جبری در ریاضیات مدرن با تأکید بر ایجاد تصویری منسجم از نحوه تعامل همه آنها ارائه می دهد. علاوه بر مطالب استاندارد در مورد گروهها، حلقهها، ماژولها، میدانها و نظریه گالوا، این کتاب شامل بحثهایی درباره موضوعات مهم دیگر، از جمله گروههای خطی، نمایشهای گروهی، حلقههای آرتینی، مدولهای پروژکتوری، تزریقی و مسطح، حوزههای Dedekind و مرکزی است. جبرهای ساده همه قضایای مهم معمولاً بدون اثبات، اما اغلب با بحث در مورد ایدههای شهودی پشت آن اثباتها مورد بحث قرار میگیرند. این راهنمای روشنگر هم برای دانشجویان فارغ التحصیل در رشته ریاضیات که تحصیلات خود را شروع می کنند و هم برای محققانی که مایلند تصویر بزرگتر ساختارهای جبری را که با آنها روبرو می شوند درک کنند ایده آل است.
Algebraic structures have come to be ubiquitous in mathematics, with almost all mathematicians encountering groups, rings, fields or more exotic related objects during the course of their research. This book presents an overview of some of the most important algebraic structures in modern mathematics, with an emphasis on creating a coherent picture of how they all interact. In addition to the standard material on groups, rings, modules, fields, and Galois theory, the book includes discussions of other important topics, including linear groups, group representations, Artinian rings, projective, injective and flat modules, Dedekind domains, and central simple algebras. All of the important theorems are discussed, typically without proofs, but often with a discussion of the intuitive ideas behind those proofs. This insightful guide is ideal for both graduate students in mathematics who are beginning their studies, and researchers who wish to understand the bigger picture of the algebraic structures they encounter.
over S Title (C) 2012 byThe Mathematical Association of America Print Edition ISBN 978-0-88385-355-9 Electronic Edition ISBN 978-1-61444-211-0 Library of Congress Catalog Card Number 2012950687 A Guide to Groups, Rings, and Fields Editors List of Published Books Contents Preface A Guide to this Guide CHAPTER 1 Algebra: Classical, Modern, and Ultramodern The Beginnings of Modern Algebra Modern Algebra Ultramodern Algebra What Next? CHAPTER 2 Categories Categories Functors Natural Transformations Products, Coproducts, and Generalizations CHAPTER 3 Algebraic Structures Structures with One Operation Rings Actions Semirings Algebras Ordered Structures CHAPTER 4 Groups and their Representations Definitions Groups and homomorphisms Subgroups Actions G acting on itself Some Important Examples Permutation groups Symmetry groups Other examples Topological groups Free groups Reframing the Definitions Orbits and Stabilizers Stabilizers Orbits Acting by multiplication Cosets Counting cosets and elements Double cosets A nice example Homomorphisms and Subgroups Kernel, image, quotient Homomorphism theorems Exact sequences Hölder's dream Many Cheerful Subgroups Generators, cyclic groups Elements of finite order Finitely generated groups and the Burnside problem Other nice subgroups Conjugation and the class equation p-groups Sylow's Theorem and Sylow subgroups Sequences of Subgroups Composition series Central series, derived series, nilpotent, solvable New Groups from Old Direct products Semidirect products Isometries of R3 Free products Direct sums of abelian groups Inverse limits and direct limits Generators and Relations Definition and examples Cayley graphs The word problem Abelian Groups Torsion The structure theorem Small Groups Order four, order p2 Order six, order pq Order eight, order p3 And so on Groups of Permutations Cycle notation and cycle structure Conjugation and cycle structure Transpositions as generators Signs and the alternating groups Transitive subgroups Automorphism group of S_n Some Linear Groups Definitions and examples Generators The regular representation Diagonal and upper triangular Normal subgroups PGL Linear groups over finite fields Representations of Finite Groups Definitions Examples Constructions Decomposing into irreducibles Direct products Characters Character tables Going through quotients Going up and down Representations of S_4 CHAPTER 5 Rings and Modules Definitions Rings Modules Torsion Bimodules Ideals Restriction of scalars Rings with few ideals More Examples, More Definitions The integers Fields and skew fields Polynomials R-algebras Matrix rings Group algebras Monsters Some examples of modules Nil and nilpotent ideals Vector spaces as K[X]-modules Q and Q/Z as Z-modules Why study modules? Homomorphisms, Submodules, and Ideals Submodules and quotients Quotient rings Irreducible modules, simple rings Small and large submodules Composing and Decomposing Direct sums and products Complements Direct and inverse limits Products of rings Free Modules Definitions and examples Vector spaces Traps Generators and free modules Homomorphisms of free modules Invariant basis number Commutative Rings, Take One Prime ideals Primary ideals The Chinese Remainder Theorem Rings of Polynomials Degree The evaluation homomorphism Integrality Unique factorization and ideals Derivatives Symmetric polynomials and functions Polynomials as functions The Radical, Local Rings, and Nakayama's Lemma The Jacobson radical Local rings Nakayama's Lemma Commutative Rings: Localization Localization Fields of fractions An important example Modules under localization Ideals under localization Integrality under localization Localization at a prime ideal What if R is not commutative? Hom Making Hom a module Functoriality Additivity Exactness Tensor Products Definition and examples Examples Additivity and exactness Over which ring? When R is commutative Extension of scalars, aka base change Extension and restriction Tensor products and Hom Finite free modules Tensoring a module with itself Tensoring two rings Projective, Injective, Flat Projective modules Injective modules Flat modules If R is commutative Finiteness Conditions for Modules Finitely generated and finitely cogenerated Artinian and Noetherian Finite length Semisimple Modules Definitions Basic properties Socle and radical Finiteness conditions Structure of Rings Finiteness conditions for rings Simple Artinian rings Semisimple rings Artinian rings Non-Artinian rings Factorization in Domains Units, irreducibles, and the rest Existence of factorization Uniqueness of factorization Principal ideal domains Euclidean domains Greatest common divisor Dedekind domains Finitely Generated Modules over Dedekind Domains The structure theorems Application to abelian groups Application to linear transformations CHAPTER 6 Fields and Skew Fields Fields and Algebras Some examples Characteristic and prime fields K-algebras and extensions Two kinds of K-homomorphisms Generating sets Compositum Linear disjointness Algebraic Extensions Definitions Transitivity Working without an A Norm and trace Algebraic elements and homomorphisms Splitting fields Algebraic closure Finite Fields Transcendental Extensions Transcendence basis Geometric examples Noether Normalization Luroth's Theorem Symmetric functions Separability Separable and inseparable polynomials Separable extensions Separability and tensor products Norm and trace Purely inseparable extensions Separable closure Primitive elements Automorphisms and Normal Extensions Automorphisms Normal extensions Galois Theory Galois extensions and Galois groups The Galois group as topological group The Galois correspondence Composita Norm and trace Normal bases Solution by radicals Determining Galois groups The inverse Galois problem Analogies and generalizations Skew Fields and Central Simple Algebras Definition and basic results Quaternion algebras Skew fields over R Tensor products Splitting fields Reduced norms and traces The Skolem-Noether Theorem The Brauer group Bibliography Index of Notations Index About the Author