##
**A course in algebra. Transl. from the Russian by Alexander Retakh.**
*(English)*
Zbl 1016.00003

Graduate Studies in Mathematics. 56. Providence, RI: American Mathematical Society (AMS). ix, 511 p. (2003).

This textbook on general algebra grew out of the various courses taught by the author, over several years, at Moscow State University and at The Independent University of Moscow. Ernest B. Vinberg, well-known as a prominent researcher in algebra and geometry, author of several monographs and encyclopedic articles, and as an experienced university teacher, presents here his variant of a two-year algebra course at the upper level as it is taught at Russian top-universities.

Indeed, this is a rather comprehensive text on modern algebra written for advanced undergraduate and basic graduate algebra classes. According to the author’s expertise and methodological mastery, this textbook comes with many particular features, individual originalities, instructive teaching examples, and other interesting extras.

The perhaps most manifest feature of the book is that the author has managed to avoid any technically involved proofs, without loosing too much of mathematical rigor. Instead, he has tried to replace tedious calculations and difficult deductions by more conceptual proofs, again without becoming too abstract or overstraining. Other features of the book are provided by a number of subjects included here, which are usually considered of being beyond a regular algebra course (e.g., finitely generated algebras and basic affine algebraic geometry, basic invariant theory, Lie groups and their representations, Lie algebras and the adjoint representation, etc.), and by the rather individual but very functional arrangement of the material, which avoids any redundance and allows to cover a plenty of important topics.

As to the precise contents, there are twelve chapters covering the following subjects:

Chapter 1 gives an introduction to the basic algebraic structures, including abelian groups, rings and fields, substructures, the field of complex numbers, \(\mathbb{Z}\) and \(\mathbb{Z}/n\mathbb{Z}\), the concept of vector space, algebras, and matrix algebras.

Chapter 2 explains the elements of linear algebra, that is systems of linear equations, bases and dimension of a finitely generated vector space, linear maps, and determinants.

Chapter 3 is devoted to polynomial algebras in one and several variables, including the fundamental theorem of algebra, the Cartesian sign rule for real polynomials, factorization in Euclidean domains, the main theorem on symmetric polynomials in several variables, and Cardano’s formula for the zeros of the cubic equation.

Chapter 4 treats the usual facts from elementary group theory, with a very nice section on groups in geometry and physics, and ends with the homomorphism theorem for group homomorphisms.

Chapter 5 returns to linear algebra and covers functionals, bilinear and quadratic forms as well as Euclidean and Hermitian spaces.

Chapter 6 discusses then linear operators and their eigenvalues, ending up with the Jordan normal form and its application to systems of linear ordinary differential equations with constant coefficients.

Chapter 7 provides the basics of affine and projective geometry, with a special emphasis on convex sets, affine transformations, affine quadrics, and elementary projective geometry.

Chapter 8 presents the essentials of multilinear algebra and its applications. This incorporates the tensor algebra, the symmetric algebra, and the exterior algebra of a vector space. The Plücker embedding of the Grassmannian and Pfaffian forms appears as a useful application.

Chapter 9 is entitled “Commutative Algebra” and deals with ideals in rings, modules over principal domains, Noetherian rings, algebraic field extensions, finitely generated algebras and affine algebraic sets, and the prime factorization in commutative rings (factoriality).

Chapter 10 returns to group theory and does the more advanced things, such as direct and semidirect products, group actions on sets, the Sylow structure theorems for finite groups, simple groups, solvable groups, Galois extensions of fields, up to the fundamental theorem of Galois theory.

Chapter 11 is devoted linear representations of groups and associative algebras. The author discusses the complete reducibility of linear representations of finite and compact groups, the description of finite-dimensional associative algebras over a field, semisimple algebras, the structure of representations of finite groups, the basic concepts of invariant theory for actions of finite and compact groups, concluding with the structure of division algebras, the Frobenius theorem, and Wedderburn’s theorem.

Chapter 12, the final chapter of the book, gives an introduction to Lie groups and their basic properties. This includes the structure of the exponential map, the tangent Lie algebra and the adjoint representation, linear representations of Lie groups, and reductive Lie groups.

Thus as one can see, E. Vinberg has touched upon quite a variety of topics, both basic and more advanced ones, and he has covered a remarkable wealth of material. Apart from having always stressed the link to other mathematical disciplines, like number theory and geometry, and to related problems in physics, the author has enriched this text by a huge number of illustrating examples which appear as just as helpful and valuable as the related theorems. These examples are utmost carefully and skillfully selected and placed, which once more provides striking evidence of the author’s outstanding teaching experience. Also there are numerous exercises scattered over the entire text, together with answers to a selected number of them at the end of the book.

All in all, this is a masterly textbook on basic algebra. It is, at the same time, demanding and down-to-earth, challenging and user-friendly, abstract and concrete, concise and comprehensible, and above all extremely educating, inspiring and enlightening, thereby accessible to everyone. Yes, this book is a great advertising for mathematics!

Indeed, this is a rather comprehensive text on modern algebra written for advanced undergraduate and basic graduate algebra classes. According to the author’s expertise and methodological mastery, this textbook comes with many particular features, individual originalities, instructive teaching examples, and other interesting extras.

The perhaps most manifest feature of the book is that the author has managed to avoid any technically involved proofs, without loosing too much of mathematical rigor. Instead, he has tried to replace tedious calculations and difficult deductions by more conceptual proofs, again without becoming too abstract or overstraining. Other features of the book are provided by a number of subjects included here, which are usually considered of being beyond a regular algebra course (e.g., finitely generated algebras and basic affine algebraic geometry, basic invariant theory, Lie groups and their representations, Lie algebras and the adjoint representation, etc.), and by the rather individual but very functional arrangement of the material, which avoids any redundance and allows to cover a plenty of important topics.

As to the precise contents, there are twelve chapters covering the following subjects:

Chapter 1 gives an introduction to the basic algebraic structures, including abelian groups, rings and fields, substructures, the field of complex numbers, \(\mathbb{Z}\) and \(\mathbb{Z}/n\mathbb{Z}\), the concept of vector space, algebras, and matrix algebras.

Chapter 2 explains the elements of linear algebra, that is systems of linear equations, bases and dimension of a finitely generated vector space, linear maps, and determinants.

Chapter 3 is devoted to polynomial algebras in one and several variables, including the fundamental theorem of algebra, the Cartesian sign rule for real polynomials, factorization in Euclidean domains, the main theorem on symmetric polynomials in several variables, and Cardano’s formula for the zeros of the cubic equation.

Chapter 4 treats the usual facts from elementary group theory, with a very nice section on groups in geometry and physics, and ends with the homomorphism theorem for group homomorphisms.

Chapter 5 returns to linear algebra and covers functionals, bilinear and quadratic forms as well as Euclidean and Hermitian spaces.

Chapter 6 discusses then linear operators and their eigenvalues, ending up with the Jordan normal form and its application to systems of linear ordinary differential equations with constant coefficients.

Chapter 7 provides the basics of affine and projective geometry, with a special emphasis on convex sets, affine transformations, affine quadrics, and elementary projective geometry.

Chapter 8 presents the essentials of multilinear algebra and its applications. This incorporates the tensor algebra, the symmetric algebra, and the exterior algebra of a vector space. The Plücker embedding of the Grassmannian and Pfaffian forms appears as a useful application.

Chapter 9 is entitled “Commutative Algebra” and deals with ideals in rings, modules over principal domains, Noetherian rings, algebraic field extensions, finitely generated algebras and affine algebraic sets, and the prime factorization in commutative rings (factoriality).

Chapter 10 returns to group theory and does the more advanced things, such as direct and semidirect products, group actions on sets, the Sylow structure theorems for finite groups, simple groups, solvable groups, Galois extensions of fields, up to the fundamental theorem of Galois theory.

Chapter 11 is devoted linear representations of groups and associative algebras. The author discusses the complete reducibility of linear representations of finite and compact groups, the description of finite-dimensional associative algebras over a field, semisimple algebras, the structure of representations of finite groups, the basic concepts of invariant theory for actions of finite and compact groups, concluding with the structure of division algebras, the Frobenius theorem, and Wedderburn’s theorem.

Chapter 12, the final chapter of the book, gives an introduction to Lie groups and their basic properties. This includes the structure of the exponential map, the tangent Lie algebra and the adjoint representation, linear representations of Lie groups, and reductive Lie groups.

Thus as one can see, E. Vinberg has touched upon quite a variety of topics, both basic and more advanced ones, and he has covered a remarkable wealth of material. Apart from having always stressed the link to other mathematical disciplines, like number theory and geometry, and to related problems in physics, the author has enriched this text by a huge number of illustrating examples which appear as just as helpful and valuable as the related theorems. These examples are utmost carefully and skillfully selected and placed, which once more provides striking evidence of the author’s outstanding teaching experience. Also there are numerous exercises scattered over the entire text, together with answers to a selected number of them at the end of the book.

All in all, this is a masterly textbook on basic algebra. It is, at the same time, demanding and down-to-earth, challenging and user-friendly, abstract and concrete, concise and comprehensible, and above all extremely educating, inspiring and enlightening, thereby accessible to everyone. Yes, this book is a great advertising for mathematics!

Reviewer: Werner Kleinert (Berlin)

### MSC:

00A05 | Mathematics in general |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |