Basic Notions of Algebra I.R. Shafarevich Translated from the Russian by M. Reid Contents Preface 4 1. What is Algebra? 6 The idea of coordinatisation. Examples:“ctionary 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, examplesofUFDs. 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 fa 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 offinite type. Modules of finite type over a principal ideal domain. Noetherian modules and rings. Noetherian rings and rings offinite 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 non- commuting polynomial ring. Exterior algebra; superalgebras; Clifford algebra. Simple rmgs and algebras. Left and right ideals of the endomorplusm 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 Iemma 10. Semisimple Modules and Rings 79 Semisimplicity. A group algebra is senusimple. Modules over a semisimple ring. Semi- simple rings of firute length; Wedderburn's theorem. Simple rings of finite length and the fundamental theorem of projective geometry. FactOfS and continuous geometries. Semisirnple 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 tR or over finite fields. Tsen's theorem and quasi-algebraically closed fields. Central division algebras offinite rank over the p-adic and rational fields. 12. The Notion of a Group 96 Transformation groups, symmetries, automorphisms. Symmetries of dynamical sys- tems 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 S0(3). Zeeman effect. C. Representations of the Classical Complex Lie Groups 174 Representations of noncompact Lie groups. Comptete 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 C. Classification of Unramified Covers 182 Classification of unramified covers and the fundamental group D.Invariant Theory The first fundamental theorem ofinvarant theory E. Group Representations and the Classification of Elementary Particles 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 submarufolds 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 coho- mology 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 4Vec(X). Periodicity and the functors K,(X). Ki(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, Ki and K of a ring. K2 0f 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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