本书论述代数学及其在现代数学和科学中的地位，高度原创且内容充实。作者通过讨论大学代数课程，如李群、上同调、范畴论等，阐述每个代数概念的起源与物理现象及其他数学分支之间的联系。

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