Introduction to Algebra. Questo articolo non è disponibile.Introduction to Algebra

This textbook, written by a dedicated and successful pedagogue who developed the present undergraduate algebra course at Moscow State University, differs in several respects from other algebra textbooks available in English. The book reflects the Soviet approach to teaching mathematics with its emphasis on applications and problem-solving -- note that the mathematics department in Moscow is called the I~echanics-Mathematics" Faculty. In the first place, Kostrikin's textbook motivates many of the algebraic concepts by practical examples, for instance, the heated plate problem used to introduce linear equations in Chapter 1. In the second place, there are a large number of exercises, so that the student can convert a vague passive understanding to active mastery of the new ideas. Thes~ problems are intended to be challenging but doable by the student; the harder ones have hints at the back of the book. This feature also makes the book ideally suited for learning algebra on one's own outside of the framework of an organized course. In the third place, the author treats material which is usually not part of an elementary course but which is fundamental in applications. Thus, Part II includes an introduction to the classical groups and to representation theory. With many American colleges now trying to bring their undergraduate mathematics curriculum closer to applications, it seems worthwhile to translate Soviet textbooks which reflect their greater experience in this area of mathematical pedagogy.

I. Foundations of Algebra.- Further reading.- 1. Sources of algebra.- §1. Algebra in brief.- §2. Some model problems.- 1. Solvability of equations in radicals.- 2. The states of a molecule.- 3. Coding information.- 4. The heated plate problem.- §3. Systems of linear equations. The first steps.- 1. Terminology.- 2. Equivalence of linear systems.- 3. Reducing to step form.- 4. Studying a system of linear equations.- 5. Some remarks and examples.- §4. Determinants of small order.- Exercises.- §5. Sets and mappings.- 1. Sets.- 2. Mappings.- Exercises.- §6. Equivalence relations. Quotient maps.- 1. Binary relations.- 2. Equivalence relations.- 3. Quotient maps.- 4. Ordered sets.- Exercises.- §7. The principle of mathematical induction.- §8. Integer arithmetic.- 1. The fundamental theorem of arithmetic.- 2. g.c.d. and l.c.m. in ZZ.- 3. The division algorithm in ZZ.- Exercises.- 2. Vector spaces. Matrices.- §1. Vector spaces.- 1. Motivation.- 2. Basic definitions.- 3. Linear combinations. Linear span.- 4. Linear dependence.- 5. Bases. Dimension.- Exercises.- §2. The rank of a matrix.- 1. Back to equations.- 2. The rank of a matrix.- 3. Solvability criterion.- Exercises.- §3. Linear maps. Matrix operations.- 1. Matrices and maps.- 2. Matrix multiplication.- 3. Square matrices.- Exercises.- §4. The space of solutions.- 1. Solving a homogeneous linear system.- 2. Linear manifolds. Solving a non-homogeneous system.- 3. The rank of a product of matrices.- 4. Equivalence classes of matrices.- Exercises.- 3. Determinants.- §1. Determinants: construction and basic properties.- 1. Construction by induction.- 2. Basic properties of determinants.- Exercises.- §2. Further properties of determinants.- 1. Expanding the determinant along an arbitrary column.- 2. The properties of determinants relating to columns.- 3. The transpose determinant.- 4. Determinants of special matrices.- 5. Building up a theory of determinants.- Exercises.- §3. Applications of determinants.- 1. Criterion for a matrix to be non-singular.- 2. Computing the rank of a matrix.- Exercises.- 4. Algebraic structures (groups, rings, fields).- §1. Sets with algebraic operations.- 1. Binary operations.- 2. Semigroups and monoids.- 3. Generalized associativity; powers.- 4. Invertible elements.- Exercises.- §2. Groups.- 1. Definition and examples.- 2. Systems of generators.- 3. Cyclic groups.- 4. The symmetric group and the alternating group 153 Exercises.- §3. Morphisms of groups.- 1. Isomorphisms.- 2. Komomorphisms.- 3. Glossary. Examples.- 4. Cosets of a subgroup.- 5. The monomorphism Sn ? GN(n).- Exercises.- §4. Rings and fields.- 1. The definition and general properties of rings.- 2. Congruences. The ring of residue classes.- 3. Ring homomorphisms and ideals.- 4. The concept of quotient group and quotient ring.- 5. Types of rings. Fields.- 6. The characteristic of a field.- 7. A remark on linear systems.- Exercises.- 5. Complex numbers and polynomials.- §1. The field of complex numbers.- 1. An auxiliary construction.- 2. The complex plane.- 3. Geometrical interpretation of operations with complex numbers.- 4. Raising to powers and extracting roots.- 5. Uniqueness theorem.- Exercises.- §2. Rings of polynomials.- 1. Polynomials in one variable.- 2. Polynomials in several variables.- 3. The division algorithm.- Exercises.- §3. Factoring in polynomial rings.- 1. Elementary divisibility properties.- 2. g.c.d. and l.c.m. in rings.- 3. Unique factorization in Euclidean rings.- 4. Irreducible polynomials.- Exercises.- §4. The field of fractions.- 1. Construction of the field of fractions of an integral domain.- 2. The field of rational functions.- 3. Primary rational functions.- Exercises.- 6. Roots of polynomials.- §1. General properties of roots.- 1. Roots and linear factors.- 2. Polynomial functions.- 3. Differentiation in polynomial rings.- 4. Multiple factors.- 5. Vieta’s formulas.- Exercises.- §2. Symmetric polynomials.- 1. The ring of symmetric polynomials.- 2. The fundamental theorem on symmetric polynomials.- 3. The method of undetermined coefficients.- 4. The discriminant of a polynomial.- 5. The resultant.- Exercises.- §3. (E is algebraically closed.- 1. Statement of the fundamental theorem.- 2. The splitting field of a polynomial.- 3. Proof of the Fundamental Theorem.- §4. Polynomials with real coefficients.- 1. Factorization in IR[X].- 2. The problem of isolating the roots of a polynomial.- 3. Stable polynomials.- Exercises.- II. Groups, Rings, Modules.- Further reading.- 7. Groups.- §1. Classical groups in low dimensions.- 1. General definitions.- 2. Parametrization of SU(2) and SO(3).- 3. The epimorphism SU(2) ? SO(3).- 4. Geometrical characterization of SO(3).- Exercises.- §2. Group actions on sets.- 1. Homomorphisms G ? S(ft).- 2. The orbit and stationary subgroup of a point.- 3. Examples of group actions on sets.- 4. Homogeneous spaces.- Exercises.- §3. Some group theoretic constructions.- 1. General theorems on group homomorphisms.- 2. Solvable groups.- 3. Simple groups.- 4. Products of groups.- 5. Generators and defining relations.- Exercises.- §4. The Sylow theorems.- Exercises.- §5. Finite abelian groups.- 1. Primary abelian groups.- 2. The structure theorem for finite abelian groups 381 Exercises.- 8. Elements of representation theory.- §1. Definitions and examples of linear representations.- 1. Basic concepts.- 2. Examples of linear representations.- Exercises.- §2. Unitary and reducible representations.- 1. Unitary representations.- 2. Complete reducibility.- Exercises.- §3. Finite rotation groups.- 1. The orders of finite subgroups of SO(3).- 2. Symmetry groups for regular polyhedra.- Exercises.- §4. Characters of linear representations.- 1. Schur’s lemma and corollary.- 2. Characters of representations.- Exercises.- §5. Irreducible representations of finite groups.- 1. The number of irreducible representations.- 2. The degrees of the irreducible representations.- 3. Representations of abelian groups.- 4. Representations of certain special groups.- Exercises.- §6. Representations of SU(2) and SO(3).- Exercises.- §7. Tensor products of representations.- 1. The dual representation.- 2. Tensor products of representations.- 3. The ring of characters.- 4. Invariants of linear groups.- Exercises.- 9. Toward a theory of fields, rings and modules.- §1. Finite field extensions.- 1. Primitive elements and the degree of an extension.- 2. Isomorphism of splitting fields.- 3. Finite fields.- 4. The Mobius inversion formula and its applications.- Exercises.- §2. Various results about rings.- 1. More examples of unique factorization domains.- 2. Ring theoretic constructions.- 3. Number theoretic applications.- Exercises.- §3. Modules.- 1. Basic facts about modules.- 2. Free modules.- 3. Integral elements of a ring.- 4. Unimodular sequences of polynomials.- §4. Algebras over a field.- 1. Definitions and examples of algebras.- 2. Division rings (skew fields).- 3. Group algebras and modules over them.- 4. Non-associative algebras.- Exercises.- Appendix. The Jordan normal form of a matrix.- Hints to the exercises.