# Introduction to abstract algebra

##### By: Smith, Jonathan D.H.

Series: Textbooks in Mathematics. Publisher: Florida CRC Press 2016Edition: 2nd ed.Description: xii, 340 p.ISBN: 9781498731614.Subject(s): Hardy spaces | Quasi-metric spacesDDC classification: 512.02 Summary: 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. https://www.crcpress.com/Introduction-to-Abstract-Algebra-Second-Edition/Smith/p/book/9781498731614Item type | Current location | Item location | Collection | Call number | Status | Date due | Barcode |
---|---|---|---|---|---|---|---|

Books | Vikram Sarabhai Library | Slot 1361 (0 Floor, East Wing) | Non-fiction | 512.02 S6I6 (Browse shelf) | Available | 192884 |

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

Functions

Specifying functions

Composite functions

Linear functions

Semigroups of functions

Injectivity and surjectivity

Isomorphisms

Groups of permutations

Exercises

Study projects

Notes

Summary

Equivalence

Kernel and equivalence relations

Equivalence classes

Rational numbers

The First Isomorphism Theorem for Sets

Modular arithmetic

Exercises

Study projects

Notes

Groups and Monoids

Semigroups

Monoids

Groups

Componentwise structure

Powers

Submonoids and subgroups

Cosets

Multiplication tables

Exercises

Study projects

Notes

Homomorphisms

Normal subgroups

Quotients

The First Isomorphism Theorem for Groups

The Law of Exponents

Cayley’s Theorem

Exercises

Study projects

Notes

Rings

Rings

Distributivity

Subrings

Ring homomorphisms

Ideals

Quotient rings

Polynomial rings

Substitution

Exercises

Study projects

Notes

Fields

Integral domains

Degrees

Fields

Polynomials over fields

Principal ideal domains

Irreducible polynomials

Lagrange interpolation

Fields of fractions

Exercises

Study projects

Notes

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

Modules

Endomorphisms

Representing a ring

Modules

Submodules

Direct sums

Free modules

Vector spaces

Abelian groups

Exercises

Study projects

Notes

Group Actions

Actions

Orbits

Transitive actions

Fixed points

Faithful actions

Cores

Alternating groups

Sylow Theorems

Exercises

Study projects

Notes

Quasigroups

Quasigroups

Latin squares

Division

Quasigroup homomorphisms

Quasigroup homotopies

Principal isotopy

Loops

Exercises

Study projects

Note

Index

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.

https://www.crcpress.com/Introduction-to-Abstract-Algebra-Second-Edition/Smith/p/book/9781498731614

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