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**Introduction to algebra. Part 1: Basic algebra. Textbook.
(Введение в алгебру. Част’ 1: Основы алгебры. Учебное пособие.)**
*(Russian)*
Zbl 0954.15001

Moskva: FIZMATLIT, Fiziko-Matematicheskaya Literatura. 271 p. (2000).

The book is Volume 1 of the trilogy (Volume 2: Linear algebra (2000; reviewed in Zbl 0954.15002 below), Volume 3: Basic algebraic structures see Zbl 0969.08001) (Remark of the reviewer: i.e. Algebraic systems)), written by the prominent Russian mathematician A. I. Kostrikin. The trilogy is intended for undergraduates with majors in mathematics and applied mathematics.

The book covers the first semester course of Algebra, delivered at the Moscow State University. Its main aim is to work up for the beginners the approach in dealing with algebra, linear algebra and geometry. It consists of six chapters (1. Sources of algebra; 2. Matrices; 3. Determinants; 4. Groups, rings, fields; 5. Complex numbers and polynomials; 6. Roots of polynomials) and an appendix: Unsolved problems on polynomials.

Chapter 1 consists of the basic concepts and the notions of algebra, including the list of the key factors for forming of modern algebra, model problems, linear systems, determinants, sets and mappings, binary relations (equivalence and orderings), the induction principle, permutations and integers.

In Chapter 2 on the base of a pure algebraic approach the fundamentals of the theory of matrices are developed systematically via the notions of a vector, a vector space, linear dependence, the rank of a linear system. The links between matrices and linear mappings are established.

Chapter 3 deals with determinants and their basic applications.

Chapter 4 introduces fundamental algebraic systems, namely, groups, rings and fields. Basic properties of these systems are characterized in terms of different types of morphisms.

In Chapter 5 the field of complex numbers and rings of polynomials are studied systematically via the notions of divisibility, integral and Euclidean domains, irreducible and primitive polynomials, field of quotients.

Chapter 6 deals with roots of polynomials. The ring of symmetric polynomials, algebraic completeness of the field of complex numbers and polynomials with real coefficients are studied systematically.

The Appendix consists of formulations and comments on Jacobi’s problem, the discriminant problem, the problem of two generative elements for the ring of polynomials, the problem of critical points and values, the problem of global convergence for Newton’s method.

The distinguishing features of the book are the following ones: 1) clearness, clarity and compactness of exposition; 2) the concentric style of presentation; 3) a variety of skilfully selected examples (from very simple to very complex ones).

The book is of exceptional interest for the following categories of potential readers: 1) students and postgraduates (in the capacity of the basic textbook and a handbook); 2) lecturers delivering courses of Algebra at universities (in the capacity of the source for developing different courses); 3) researchers (in the capacity of a handbook).

The book covers the first semester course of Algebra, delivered at the Moscow State University. Its main aim is to work up for the beginners the approach in dealing with algebra, linear algebra and geometry. It consists of six chapters (1. Sources of algebra; 2. Matrices; 3. Determinants; 4. Groups, rings, fields; 5. Complex numbers and polynomials; 6. Roots of polynomials) and an appendix: Unsolved problems on polynomials.

Chapter 1 consists of the basic concepts and the notions of algebra, including the list of the key factors for forming of modern algebra, model problems, linear systems, determinants, sets and mappings, binary relations (equivalence and orderings), the induction principle, permutations and integers.

In Chapter 2 on the base of a pure algebraic approach the fundamentals of the theory of matrices are developed systematically via the notions of a vector, a vector space, linear dependence, the rank of a linear system. The links between matrices and linear mappings are established.

Chapter 3 deals with determinants and their basic applications.

Chapter 4 introduces fundamental algebraic systems, namely, groups, rings and fields. Basic properties of these systems are characterized in terms of different types of morphisms.

In Chapter 5 the field of complex numbers and rings of polynomials are studied systematically via the notions of divisibility, integral and Euclidean domains, irreducible and primitive polynomials, field of quotients.

Chapter 6 deals with roots of polynomials. The ring of symmetric polynomials, algebraic completeness of the field of complex numbers and polynomials with real coefficients are studied systematically.

The Appendix consists of formulations and comments on Jacobi’s problem, the discriminant problem, the problem of two generative elements for the ring of polynomials, the problem of critical points and values, the problem of global convergence for Newton’s method.

The distinguishing features of the book are the following ones: 1) clearness, clarity and compactness of exposition; 2) the concentric style of presentation; 3) a variety of skilfully selected examples (from very simple to very complex ones).

The book is of exceptional interest for the following categories of potential readers: 1) students and postgraduates (in the capacity of the basic textbook and a handbook); 2) lecturers delivering courses of Algebra at universities (in the capacity of the source for developing different courses); 3) researchers (in the capacity of a handbook).

Reviewer: Vladimir G.Skobelev (Donetsk)

### MSC:

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

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

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

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

00A05 | Mathematics in general |