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020 _a9783031500626
_9978-3-031-50062-6
024 7 _a10.1007/978-3-031-50062-6
_2doi
050 4 _aQA150-272
072 7 _aPBF
_2bicssc
072 7 _aMAT002000
_2bisacsh
072 7 _aPBF
_2thema
100 1 _aBroué, Michel.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_970147
245 1 0 _aFrom Rings and Modules to Hopf Algebras
_h[electronic resource] :
_bOne Flew Over the Algebraist's Nest /
_cby Michel Broué.
250 _a1st ed. 2024.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2024.
300 _aX, 533 p. 20 illus., 10 illus. in color.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
505 0 _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.
520 _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.
650 0 _aAlgebra.
_91501
650 0 _aAlgebraic fields.
_970148
650 0 _aPolynomials.
_964269
650 0 _aGroup theory.
_970149
650 0 _aAlgebra, Homological.
_970150
650 0 _aCommutative algebra.
_970151
650 0 _aCommutative rings.
_970152
650 1 4 _aAlgebra.
_91501
650 2 4 _aField Theory and Polynomials.
_970153
650 2 4 _aGroup Theory and Generalizations.
_970154
650 2 4 _aCategory Theory, Homological Algebra.
_970155
650 2 4 _aCommutative Rings and Algebras.
_970156
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031500619
776 0 8 _iPrinted edition:
_z9783031500633
856 4 0 _uhttps://doi.org/10.1007/978-3-031-50062-6
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
999 _c57372
_d57372