Introduction to Abstract Algebra, Second Edition

  • Explores abstract algebra as the main tool underlying discrete mathematics and the digital world
  • Presents the fundamentals of abstract algebra, before offering deeper coverage of group and ring theory
  • Uses semigroups and monoids as stepping stones to present the concepts of groups and rings
  • Contains numerous 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
  • Offers numerous updates based on the feedback of first-edition adopters

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

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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.

Numbers
Ordering numbersThe Well-Ordering PrincipleDivisibilityThe Division AlgorithmGreatest common divisorsThe Euclidean AlgorithmPrimes and irreduciblesThe Fundamental Theorem of ArithmeticExercisesStudy projectsNotes

FunctionsSpecifying functions

Composite functionsLinear functionsSemigroups of functionsInjectivity and surjectivityIsomorphismsGroups of permutationsExercisesStudy projectsNotesSummary

EquivalenceKernel and equivalence relations

Equivalence classesRational numbersThe First Isomorphism Theorem for SetsModular arithmeticExercisesStudy projectsNotes

Groups and Monoids Semigroups

MonoidsGroupsComponentwise structurePowersSubmonoids and subgroups CosetsMultiplication tablesExercisesStudy projectsNotes

Homomorphisms Homomorphisms

Normal subgroupsQuotientsThe First Isomorphism Theorem for GroupsThe Law of ExponentsCayley’s TheoremExercisesStudy projectsNotes

Rings
Rings

DistributivitySubringsRing homomorphismsIdealsQuotient ringsPolynomial ringsSubstitutionExercisesStudy projectsNotes

Fields Integral domains

DegreesFieldsPolynomials over fieldsPrincipal ideal domains Irreducible polynomialsLagrange interpolationFields of fractionsExercisesStudy projectsNotes

FactorizationFactorization in integral domains

Noetherian domainsUnique factorization domainsRoots of polynomialsSplitting fieldsUniqueness of splitting fieldsStructure of finite fieldsGalois fieldsExercisesStudy projects Notes

ModulesEndomorphisms

Representing a ring ModulesSubmodulesDirect sumsFree modulesVector spacesAbelian groupsExercisesStudy projectsNotes

Group ActionsActions

OrbitsTransitive actionsFixed pointsFaithful actionsCoresAlternating groupsSylow TheoremsExercisesStudy projects Notes


Quasigroups
Quasigroups

Latin squaresDivisionQuasigroup homomorphismsQuasigroup homotopiesPrincipal isotopyLoopsExercisesStudy projectsNote


Index


Category: Abstract

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