Algebra I : Basic notions of algebra / A. I. Kostrikin, I. R. Shafarevich (eds.).
Idioma: Inglés Lenguaje original: Ruso Series Encyclopaedia of mathematical sciences ; v. 11Editor: Berlin : Springer-Verlag, c1990Descripción: 258 p. : il. ; 24 cmISBN: 3540170065; 0387170065 (New York)Otro título: Basic notions of algebra | Algebra 1Trabajos contenidos: Shafarevich, I. R. 1923- Basic notions of algebraOtra clasificación: 00A05 (12-01 13-01 16-01 18-01 20-01)Basic notions of algebra / I. R. Shafarevich -- Contents Preface [4] § 1. What is Algebra? [6] The idea of coordinatisation. Examples: dictionary of quantum mechanics and coordinatisation of finite models of incidence axioms and parallelism. §2 Fields [11] Field axioms, isomorphisms. Field of rational functions in independent variables; function field of a plane algebraic curve. Field of Laurent series and formal Laurent series. §3. Commutative Rings [17] Ring axioms: zerodivisors and integral domains. Field of fractions. Polynomial rings. Ring of polynomial functions on a plane algebraic curve. Ring of power series and formal power series. Boolean rings. Direct sums of rings. Ring of continuous functions. Factorisation; unique factorisation domains, examples of UFDs. §4. Homomorphisms and Ideals [24] Homomorphisms, ideals, quotient rings. The homomorphisms theorem. The restriction homomorphism in rings of functions. Principal ideal domains; relations with UFDs. Product of ideals. Characteristic of a field. Extension in which a given polynomial has a root Algebraically closed fields. Finite fields. Representing elements of a general ring as functions on maximal and prime ideals. Integers as functions. Ultraproducts and nonstandard analysis. Commuting differential operators. §5. Modules [33] Direct sums and free modules. Tensor products. Tensor, symmetric and exterior powers of a module, the dual module. Equivalent ideals and isomorphism of modules. Modules of differential forms and vector fields. Families of vector spaces and modules. §6. Algebraic Aspects of Dimension [41] Rank of a module. Modules of finite type. Modules of finite type over a principal ideal domain. Noe the nan modules and rings. Noetherian rings and rings of finite type. The case of graded rings. Transcendence degree of an extension. Finite extensions. §7. The Algebraic View of Infinitesimal Notions [50] Functions modulo second order infinitesimals and the tangent space of a manifold. Singular points. Vector fields and first order differential operators. Higher order infinitesimals. Jets and differential operators. Completions of rings, p-adic numbers. Normed fields. Valuations of the fields of rational numbers and rational functions. The p-adic number fields in number theory. § 8. NoncommutatiVe Rings [61] Basic definitions. Algebras over rings. Ring of endomorphisms of a module. Group algebra. Quaternions and division algebras. Twistor fibration. Endomorphisms of n-dimensional vector space over a division algebra. Tensor algebra and the noncommuting polynomial ring. Exterior algebra; superalgebras; Clifford algebra. Simple rings and algebras. Left and right ideals of the endomorphism ring of a vector space over a division algebra. §9. Modules over Noncommutative Rings [74] Modules and representations. Representations of algebras in matrix form. Simple modules, composition series, the Jordan-Holder theorem. Length of a ring or module. Endomorphisms of a module. Schur’s lemma § 10. Semisimple Modules and Rings [79] Semisimplicity. A group algebra is semisimple. Modules over a semisimple ring. Semisimple rings of finite length; Wedderburn’s theorem. Simple rings of finite length and the fundamental theorem of projective geometry. Factors and continuous geometries. Semisimple algebras of finite rank over an algebraically closed field. Applications to representations of finite groups. §11. Division Algebras of Finite Rank [90] Division algebras of finite rank over R or over finite fields. Tsen’s theorem and quasi-algebraically closed fields. Central division algebras of finite rank over the p-adic and rational fields. §12. The Notion of a Group [96] Transformation groups, symmetries, automorphisms. Symmetries of dynamical systems and conservation laws. Symmetries of physical laws. Groups, the regular action. Subgroups, normal subgroups, quotient groups. Order of an element. The ideal class group. Group of extensions of a module. Brauer group. Direct product of two groups. § 13. Examples of Groups: Finite Groups [108] Symmetric and alternating groups. Symmetry groups of regular polygons and regular polyhedrons. Symmetry groups of lattices. Crystallographic classes. Finite groups generated by reflections. § 14. Examples of Groups: Infinite Discrete Groups [124] Discrete transformation groups. Crystallographic groups. Discrete groups of motion of the Lobachevsky plane. The modular group. Free groups. Specifying a group by generators and relations. Logical problems. The fundamental group. Group of a knot. Braid group. § 15. Examples of Groups: Lie Groups and Algebraic Groups [140] Lie groups. Toruses. Their role in Liouville’s theorem. A. Compact Lie Groups [143] The classical compact groups and some of the relations between them. B. Complex Analytic Lie Groups [147] The classical complex Lie groups. Some other Lie groups. The Lorentz group. C. Algebraic Groups [150] Algebraic groups, the adele group. Tamagawa number. § 16. General Results of Group Theory [151] Direct products. The Wedderburn-Remak-Shmidt theorem. Composition series, the Jordan-Holder theorem. Simple groups, solvable groups. Simple compact Lie groups. Simple complex Lie groups. Simple finite groups, classification. § 17. Group Representations [160] A. Representations of Finite Groups [163] Representations. Orthogonality relations. B. Representations of Compact Lie Groups [167] Representations of compact groups. Integrating over a group. Helmholtz-Lie theory. Characters of compact Abelian groups and Fourier series. Weyl and Ricci tensors in 4-dimensional Riemannian geometry. Representations of SU(2) and SO(3). Zeeman effect. C. Representations of the Classical Complex Lie Groups [174] Representations of noncompact Lie groups. Complete irreducibility of representations of finite-dimensional classical complex Lie groups. § 18. Some Applications of Groups [177] A. Galois Theory [177] Galois theory. Solving equations by radicals. B. The Galois Theory of Linear Differential Equations (Picard- Vessiot Theory) [181] C. Classification of Unramified Covers [182] Classification of unramified covers and the fundamental group D. Invariant Theory [183] The first fundamental theorem of invariant theory E. Group Representations and the Classification of Elementary Particles [185] § 19. Lie Algebras and Nonassociative Algebra [188] A. Lie Algebras [188] Poisson brackets as an example of a Lie algebra. Lie rings and Lie algebras. B. Lie Theory [192] Lie algebra of a Lie group. C. Applications of Lie Algebras [197] Lie groups and rigid body motion. D. Other Nonassociative Algebras [199] The Cayley numbers. Almost complex structure on 6-dimensional submanifolds of 8-space. Nonassociative real division algebras. §20. Categories [202] Diagrams and categories. Universal mapping problems. Functors. Functors arising in topology: loop spaces, suspension. Group objects in categories. Homotopy groups. §21. Homological Algebra [213] A. Topological Origins of the Notions of Homological Algebra ... 213 Complexes and their homology. Homology and cohomology of polyhedrons. Fixed point theorem. Differential forms and de Rham cohomology, de Rham’s theorem. Long exact cohomology sequence. B. Cohomology of Modules and Groups [219] Cohomology of modules. Group cohomology. Topological meaning of the cohomology of discrete groups. C. Sheaf Cohomology [225] Sheaves; sheaf cohomology. Finiteness theorems. Riemann-Roch theorem. §22. K-theory [230] A. Topological K-theory [230] Vector bundles and the functor ~Vec(X). Periodicity and the functors Kn(X). K 1 (X) and the infinite-dimensional linear group. The symbol of an elliptic differential operator. The index theorem. B. Algebraic K-theory [234] The group of classes of projective modules. Ko, K1 and Kn of a ring. K2 of a field and its relations with the Brauer group. K-theory and arithmetic. Comments on the Literature [239] References [244] Index of Names [249] Subject Index [251]
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| 00A05 Y73 Lectures on fundamental concepts of algebra and geometry / | 00A05A Ab116 Elementos de álgebra / | 00A05A Ab116 Elementos de álgebra / | 00A05A Al394 Algebra I : Basic notions of algebra / | 00A05A Ar791 Algebra / | 00A05A Ay985 Schaum's outline of theory and problems of modern algebra / | 00A05A B261 Introduction to abstract algebra / |
Traducido de la edición original rusa (Moscú : VINITI, 1986).
Incluye referencias bibliográficas (p. 244-248) e índices.
Basic notions of algebra / I. R. Shafarevich --
Contents --
Preface [4] --
§ 1. What is Algebra? [6] --
The idea of coordinatisation. Examples: dictionary of quantum mechanics and coordinatisation of finite models of incidence axioms and parallelism. --
§2 Fields [11] --
Field axioms, isomorphisms. Field of rational functions in independent variables; function field of a plane algebraic curve. Field of Laurent series and formal Laurent series. --
§3. Commutative Rings [17] --
Ring axioms: zerodivisors and integral domains. Field of fractions. Polynomial rings. Ring of polynomial functions on a plane algebraic curve. Ring of power series and formal power series. Boolean rings. Direct sums of rings. Ring of continuous functions. Factorisation; unique factorisation domains, examples of UFDs. --
§4. Homomorphisms and Ideals [24] --
Homomorphisms, ideals, quotient rings. The homomorphisms theorem. The restriction homomorphism in rings of functions. Principal ideal domains; relations with UFDs. Product of ideals. Characteristic of a field. Extension in which a given polynomial has a root Algebraically closed fields. Finite fields. Representing elements of a general ring as functions on maximal and prime ideals. Integers as functions. Ultraproducts and nonstandard analysis. Commuting differential operators. --
§5. Modules [33] --
Direct sums and free modules. Tensor products. Tensor, symmetric and exterior powers of a module, the dual module. Equivalent ideals and isomorphism of modules. Modules of differential forms and vector fields. Families of vector spaces and modules. --
§6. Algebraic Aspects of Dimension [41] --
Rank of a module. Modules of finite type. Modules of finite type over a principal ideal domain. Noe the nan modules and rings. Noetherian rings and rings of finite type. The case of graded rings. Transcendence degree of an extension. Finite extensions. --
§7. The Algebraic View of Infinitesimal Notions [50] --
Functions modulo second order infinitesimals and the tangent space of a manifold. Singular points. Vector fields and first order differential operators. Higher order infinitesimals. Jets and differential operators. Completions of rings, p-adic numbers. Normed fields. Valuations of the fields of rational numbers and rational functions. The p-adic number fields in number theory. --
§ 8. NoncommutatiVe Rings [61] --
Basic definitions. Algebras over rings. Ring of endomorphisms of a module. Group algebra. Quaternions and division algebras. Twistor fibration. Endomorphisms of n-dimensional vector space over a division algebra. Tensor algebra and the noncommuting polynomial ring. Exterior algebra; superalgebras; Clifford algebra. Simple rings and algebras. Left and right ideals of the endomorphism ring of a vector space over a division algebra. --
§9. Modules over Noncommutative Rings [74] --
Modules and representations. Representations of algebras in matrix form. Simple modules, composition series, the Jordan-Holder theorem. Length of a ring or module. Endomorphisms of a module. Schur’s lemma --
§ 10. Semisimple Modules and Rings [79] --
Semisimplicity. A group algebra is semisimple. Modules over a semisimple ring. Semisimple rings of finite length; Wedderburn’s theorem. Simple rings of finite length and the fundamental theorem of projective geometry. Factors and continuous geometries. Semisimple algebras of finite rank over an algebraically closed field. Applications to representations of finite groups. --
§11. Division Algebras of Finite Rank [90] --
Division algebras of finite rank over R or over finite fields. Tsen’s theorem and quasi-algebraically closed fields. Central division algebras of finite rank over the p-adic and rational fields. --
§12. The Notion of a Group [96] --
Transformation groups, symmetries, automorphisms. Symmetries of dynamical systems and conservation laws. Symmetries of physical laws. Groups, the regular action. Subgroups, normal subgroups, quotient groups. Order of an element. The ideal class group. Group of extensions of a module. Brauer group. Direct product of two groups. --
§ 13. Examples of Groups: Finite Groups [108] --
Symmetric and alternating groups. Symmetry groups of regular polygons and regular polyhedrons. Symmetry groups of lattices. Crystallographic classes. Finite groups generated by reflections. --
§ 14. Examples of Groups: Infinite Discrete Groups [124] --
Discrete transformation groups. Crystallographic groups. Discrete groups of motion of the Lobachevsky plane. The modular group. Free groups. Specifying a group by generators and relations. Logical problems. The fundamental group. Group of a knot. Braid group. --
§ 15. Examples of Groups: Lie Groups and Algebraic Groups [140] --
Lie groups. Toruses. Their role in Liouville’s theorem. --
A. Compact Lie Groups [143] --
The classical compact groups and some of the relations between them. --
B. Complex Analytic Lie Groups [147] --
The classical complex Lie groups. Some other Lie groups. The Lorentz group. --
C. Algebraic Groups [150] --
Algebraic groups, the adele group. Tamagawa number. --
§ 16. General Results of Group Theory [151] --
Direct products. The Wedderburn-Remak-Shmidt theorem. Composition series, the Jordan-Holder theorem. Simple groups, solvable groups. Simple compact Lie groups. Simple complex Lie groups. Simple finite groups, classification. --
§ 17. Group Representations [160] --
A. Representations of Finite Groups [163] --
Representations. Orthogonality relations. --
B. Representations of Compact Lie Groups [167] --
Representations of compact groups. Integrating over a group. Helmholtz-Lie theory. Characters of compact Abelian groups and Fourier series. Weyl and Ricci tensors in 4-dimensional Riemannian geometry. Representations of SU(2) and SO(3). Zeeman effect. --
C. Representations of the Classical Complex Lie Groups [174] --
Representations of noncompact Lie groups. Complete irreducibility of representations --
of finite-dimensional classical complex Lie groups. --
§ 18. Some Applications of Groups [177] --
A. Galois Theory [177] --
Galois theory. Solving equations by radicals. --
B. The Galois Theory of Linear Differential Equations (Picard- --
Vessiot Theory) [181] --
C. Classification of Unramified Covers [182] --
Classification of unramified covers and the fundamental group --
D. Invariant Theory [183] --
The first fundamental theorem of invariant theory --
E. Group Representations and the Classification of Elementary --
Particles [185] --
§ 19. Lie Algebras and Nonassociative Algebra [188] --
A. Lie Algebras [188] --
Poisson brackets as an example of a Lie algebra. Lie rings and Lie algebras. --
B. Lie Theory [192] --
Lie algebra of a Lie group. --
C. Applications of Lie Algebras [197] --
Lie groups and rigid body motion. --
D. Other Nonassociative Algebras [199] --
The Cayley numbers. Almost complex structure on 6-dimensional submanifolds of 8-space. Nonassociative real division algebras. --
§20. Categories [202] --
Diagrams and categories. Universal mapping problems. Functors. Functors arising in topology: loop spaces, suspension. Group objects in categories. Homotopy groups. --
§21. Homological Algebra [213] --
A. Topological Origins of the Notions of Homological Algebra ... 213 Complexes and their homology. Homology and cohomology of polyhedrons. Fixed point theorem. Differential forms and de Rham cohomology, de Rham’s theorem. Long exact cohomology sequence. --
B. Cohomology of Modules and Groups [219] --
Cohomology of modules. Group cohomology. Topological meaning of the cohomology of discrete groups. --
C. Sheaf Cohomology [225] --
Sheaves; sheaf cohomology. Finiteness theorems. Riemann-Roch theorem. --
§22. K-theory [230] --
A. Topological K-theory [230] --
Vector bundles and the functor ~Vec(X). Periodicity and the functors Kn(X). K 1 (X) and --
the infinite-dimensional linear group. The symbol of an elliptic differential operator. --
The index theorem. --
B. Algebraic K-theory [234] --
The group of classes of projective modules. Ko, K1 and Kn of a ring. K2 of a field and its relations with the Brauer group. K-theory and arithmetic. --
Comments on the Literature [239] --
References [244] --
Index of Names [249] --
Subject Index [251] --
MR, 90k:00010
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