Introduction to Abstract Algebra Second Edition by Jonathan Smith ISBN 978-1498731614 1498731619

Introduction to Abstract Algebra Second Edition by Jonathan D. H. Smith – Ebook PDF Instant Download/Delivery:  978-1498731614, 1498731619
Full download Introduction to Abstract Algebra Second Edition  after payment

Product details: 

ISBN 10: 1498731619

ISBN 13:  978-1498731614

Author: Jonathan D. H. Smith 

Introduction to Abstract Algebra, Second Edition presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It avoids the usual groups first/rings first dilemma by introducing semigroups and monoids, the multiplicative structures of rings, along with groups.

This new edition of a widely adopted textbook covers applications from biology, science, and engineering. It offers numerous updates based on feedback from first edition adopters, as well as improved and simplified proofs of a number of important theorems. Many new exercises have been added, while new study projects examine skewfields, quaternions, and octonions.

The first three chapters of the book show how functional composition, cycle notation for permutations, and matrix notation for linear functions provide techniques for practical computation. These three chapters provide a quick introduction to algebra, sufficient to exhibit irrational numbers or to gain a taste of cryptography.

Chapters four through seven cover abstract groups and monoids, orthogonal groups, stochastic matrices, Lagrange’s theorem, groups of units of monoids, homomorphisms, rings, and integral domains. The first seven chapters provide basic coverage of abstract algebra, suitable for a one-semester or two-quarter course.

Each chapter includes exercises of varying levels of difficulty, chapter notes that point out variations in notation and approach, and study projects that cover an array of applications and developments of the theory.

The final chapters deal with slightly more advanced topics, suitable for a second-semester or third-quarter course. These chapters delve deeper into the theory of rings, fields, and groups. They discuss modules, including vector spaces and abelian groups, group theory, and quasigroups.

This textbook is suitable for use in an undergraduate course on abstract algebra for mathematics, computer science, and education majors, along with students from other STEM fields.

Table of contents: 

1. Numbers

   • Ordering numbers

   • The Well-Ordering Principle

   • Divisibility

   • The Division Algorithm

   • Greatest common divisors

   • The Euclidean Algorithm

   • Primes and irreducibles

   • The Fundamental Theorem of Arithmetic

   • Exercises

   • Study projects

   • Notes

2. Functions

   • Specifying functions

   • Composite functions

   • Linear functions

   • Semigroups of functions

   • Injectivity and surjectivity

   • Isomorphisms

   • Groups of permutations

   • Exercises

   • Study projects

   • Notes

   • Summary

3. Equivalence

   • Kernel and equivalence relations

   • Equivalence classes

   • Rational numbers

   • The First Isomorphism Theorem for Sets

   • Modular arithmetic

   • Exercises

   • Study projects

   • Notes

4. Groups and Monoids

   • Semigroups

   • Monoids

   • Groups

   • Componentwise structure

   • Powers

   • Submonoids and subgroups

   • Cosets

   • Multiplication tables

   • Exercises

   • Study projects

   • Notes

5. Homomorphisms

   • Homomorphisms

   • Normal subgroups

   • Quotients

   • The First Isomorphism Theorem for Groups

   • The Law of Exponents

   • Cayley’s Theorem

   • Exercises

   • Study projects

   • Notes

6. Rings

   • Rings

   • Distributivity

   • Subrings

   • Ring homomorphisms

   • Ideals

   • Quotient rings

   • Polynomial rings

   • Substitution

   • Exercises

   • Study projects

   • Notes

7. Fields

   • Integral domains

   • Degrees

   • Fields

   • Polynomials over fields

   • Principal ideal domains

   • Irreducible polynomials

   • Lagrange interpolation

   • Fields of fractions

   • Exercises

   • Study projects

   • Notes

8. Factorization

   • Factorization in integral domains

   • Noetherian domains

   • Unique factorization domains

   • Roots of polynomials

   • Splitting fields

   • Uniqueness of splitting fields

   • Structure of finite fields

   • Galois fields

   • Exercises

   • Study projects

   • Notes

9. Modules

   • Endomorphisms

   • Representing a ring

   • Modules

   • Submodules

   • Direct sums

   • Free modules

   • Vector spaces

   • Abelian groups

   • Exercises

   • Study projects

   • Notes

10. Group Actions

    • Actions

    • Orbits

    • Transitive actions

    • Fixed points

    • Faithful actions

    • Cores

    • Alternating groups

    • Sylow Theorems

    • Exercises

    • Study projects

    • Notes

11. Quasigroups

    • Quasigroups

    • Latin squares

    • Division

    • Quasigroup homomorphisms

    • Quasigroup homotopies

    • Principal isotopy

    • Loops

    • Exercises

    • Study projects

    • Note

People also search for:

    
w keith nicholson introduction to abstract algebra
    
mathematics 113 introduction to abstract algebra
    
roy dubisch introduction to abstract algebra
    
hungerford introduction to abstract algebra
    
a gentle introduction to abstract algebra

Tags: Jonathan Smith, Introduction to Abstract