Introduction to Algebra (häftad)
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Häftad (Paperback / softback)
Antal sidor
Softcover reprint of the original 1st ed. 1982
Springer-Verlag New York Inc.
N Koblitz
XIII, 577 p.
234 x 156 x 30 mm
822 g
Antal komponenter
1 Paperback / softback
Introduction to Algebra (häftad)

Introduction to Algebra

Häftad Engelska, 1982-05-01
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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.
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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 polyno