A guide to groups, rings, and fields [[electronic resource] /] / Fernando Q. Gouvêa

• Autore: Gouvêa Fernando Q (Fernando Quadros)
• Pubblicazione: Washington, D.C., : Mathematical Association of America, 2012
• Descrizione fisica: 1 online resource (328 p.)
• Disciplina: 512
• Soggetto topico: Algebra; Rings (Algebra); Algebraic fields
• Soggetto genere / forma: Electronic books.
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references (p. 277-283) and indexes.
• Nota di contenuto: ""copyright page ""; ""title page ""; ""Contents""; ""Preface""; ""A Guide to this Guide""; ""Algebra: Classical, Modern, and Ultramodern""; ""The Beginnings of Modern Algebra""; ""Modern Algebra""; ""Ultramodern Algebra""; ""What Next?""; ""Categories""; ""Categories""; ""Functors""; ""Natural Transformations""; ""Products, Coproducts, and Generalizations""; ""Algebraic Structures""; ""Structures with One Operation""; ""Rings""; ""Actions""; ""Semirings""; ""Algebras""; ""Ordered Structures""; ""Groups and their Representations""; ""Definitions""; ""Groups and homomorphisms""; ""Subgroups"" ; ""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""; ""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""
• Sommario/riassunto: This Guide offers a concise overview of the theory of groups, rings, and fields at the graduate level, emphasizing those aspects that are useful in other parts of mathematics. It focuses on the main ideas and how they hang together. It will be useful to both students and professionals. In addition to the standard material on groups, rings, modules, fields, and Galois theory, the book includes discussions of other important topics that are often omitted in the standard graduate course, including linear groups, group representations, the structure of Artinian rings, projective, injective and flat modules, Dedekind domains, and central simple algebras. All of the important theorems are discussed, without proofs but often with a discussion of the intuitive ideas behind those proofs. Those looking for a way to review and refresh their basic algebra will benefit from reading this Guide, and it will also serve as a ready reference for mathematicians who make use of algebra in their work.
• Titolo autorizzato: A guide to groups, rings, and fields
• ISBN: 1-61444-211-8
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910463203303321
• Lo trovi qui: Univ. Federico II
• Serie: Dolciani Mathematical Expositions