LEADER 05038nam 22007575 450 001 9910831010103321 005 20241007160752.0 010 $a3-031-50062-8 024 7 $a10.1007/978-3-031-50062-6 035 $a(MiAaPQ)EBC31097974 035 $a(Au-PeEL)EBL31097974 035 $a(CKB)30165809900041 035 $a(DE-He213)978-3-031-50062-6 035 $a(EXLCZ)9930165809900041 100 $a20240130d2024 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aFrom Rings and Modules to Hopf Algebras $eOne Flew Over the Algebraist's Nest /$fby Michel Broué 205 $a1st ed. 2024. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2024. 215 $a1 online resource (528 pages) 311 08$aPrint version: Broué, Michel From Rings and Modules to Hopf Algebras Cham : Springer International Publishing AG,c2024 9783031500619 320 $aIncludes bibliographical references and index. 327 $a1 Prerequisites and Preliminaries -- Part I Rings and Modules -- 2 Rings, Polynomials, Divisibility -- 3 Polynomial Rings in Several Indeterminates -- 4 More on Modules -- 5 On Representations of Finite Groups -- Part II Integral Domains, Polynomials, Fields -- 6 Prime and Maximal Ideals, Integral Domains -- 7 Fields, Division Rings -- Part III Finitely Generated Modules -- 8 Integrality, Noetherianity -- 9 Finitely Generated Projective Modules -- 10 Finitely Generated Modules Over Dedekind Domains -- 11 Complement on Dedekind Domains -- Part IV Characteristic Zero Linear Representations of Finite Groups -- 12 Monoidal Categories: An Introduction -- 13 Characteristic 0 Representations -- 14 Playing With the Base Field -- 15 Induction and Restriction: Some Applications to Finite Groups -- 16 Brauer?s Theorem and Some Applications -- 17 Graded Representations and Characters -- 18 The Drinfeld?Lusztig Double of a Group Algebra. 330 $aThis textbook provides an introduction to fundamental concepts of algebra at upper undergraduate to graduate level, covering the theory of rings, fields and modules, as well as the representation theory of finite groups. Throughout the book, the exposition relies on universal constructions, making systematic use of quotients and category theory ? whose language is introduced in the first chapter. The book is divided into four parts. Parts I and II cover foundations of rings and modules, field theory and generalities on finite group representations, insisting on rings of polynomials and their ideals. Part III culminates in the structure theory of finitely generated modules over Dedekind domains and its applications to abelian groups, linear maps, and foundations of algebraic number theory. Part IV is an extensive study of linear representations of finite groups over fields of characteristic zero, including graded representations and graded characters as well as a final chapter on the Drinfeld?Lusztig double of a group algebra, appearing for the first time in a textbook at this level. Based on over twenty years of teaching various aspects of algebra, mainly at the École Normale Supérieure (Paris) and at Peking University, the book reflects the audiences of the author's courses. In particular, foundations of abstract algebra, like linear algebra and elementary group theory, are assumed of the reader. Each of the of four parts can be used for a course ? with a little ad hoc complement on the language of categories. Thanks to its rich choice of topics, the book can also serve students as a reference throughout their studies, from undergraduate to advanced graduate level. 606 $aAlgebra 606 $aAlgebraic fields 606 $aPolynomials 606 $aGroup theory 606 $aAlgebra, Homological 606 $aCommutative algebra 606 $aCommutative rings 606 $aAlgebra 606 $aField Theory and Polynomials 606 $aGroup Theory and Generalizations 606 $aCategory Theory, Homological Algebra 606 $aCommutative Rings and Algebras 606 $aÀlgebra$2thub 606 $aÀlgebres de Hopf$2thub 608 $aLlibres electrònics$2thub 615 0$aAlgebra. 615 0$aAlgebraic fields. 615 0$aPolynomials. 615 0$aGroup theory. 615 0$aAlgebra, Homological. 615 0$aCommutative algebra. 615 0$aCommutative rings. 615 14$aAlgebra. 615 24$aField Theory and Polynomials. 615 24$aGroup Theory and Generalizations. 615 24$aCategory Theory, Homological Algebra. 615 24$aCommutative Rings and Algebras. 615 7$aÀlgebra 615 7$aÀlgebres de Hopf 676 $a512.9 700 $aBroue?$b Michel$00 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910831010103321 996 $aFrom Rings and Modules to Hopf Algebras$93970411 997 $aUNINA