Smith Introduction to Abstract Algebra

Erscheinungsjahr 2008
ISBN: 978-1-4200-6372-1
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 344 Seiten

Reihe: Textbooks in Mathematics

Taking a slightly different approach from similar texts, Introduction to Abstract Algebra presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It helps students fully understand groups, rings, semigroups, and monoids by rigorously building concepts from first principles. A Quick Introduction to Algebra 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. The author also uses equivalence relations to introduce rational numbers and modular arithmetic as well as to present the first isomorphism theorem at the set level. The Basics of Abstract Algebra for a First-Semester Course Subsequent chapters cover orthogonal groups, stochastic matrices, Lagrange’s theorem, and groups of units of monoids. The text also deals with homomorphisms, which lead to Cayley’s theorem of reducing abstract groups to concrete groups of permutations. It then explores rings, integral domains, and fields. Advanced Topics for a Second-Semester Course The final, mostly self-contained chapters delve deeper into the theory of rings, fields, and groups. They discuss modules (such as vector spaces and abelian groups), group theory, and quasigroups.

Zielgruppe

Advanced undergraduate students in mathematics.

Autoren/Hrsg.

Smith, Jonathan D.H.

Fachgebiete

• Mathematik | Informatik Mathematik Algebra
• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik

Inhaltsverzeichnis

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

Functions

Specifying Functions

Composite Functions

Linear Functions

Semigroups of Functions

Injectivity and Surjectivity

Isomorphisms

Groups of Permutations

Equivalence

Kernel and Equivalence Relations

Equivalence Classes

Rational Numbers

The First Isomorphism Theorem for Sets

Modular Arithmetic

Groups and Monoids

Semigroups

Monoids

Groups

Componentwise Structure

Powers

Submonoids and Subgroups

Cosets

Multiplication Tables

Homomorphisms

Homomorphisms

Normal Subgroups

Quotients

The First Isomorphism Theorem for Groups

The Law of Exponents

Cayley’s Theorem

Rings

Rings

Distributivity

Subrings

Ring Homomorphisms

Ideals

Quotient Rings

Polynomial Rings

Substitution

Fields

Integral Domains

Degrees

Fields

Polynomials over Fields

Principal Ideal Domains

Irreducible Polynomials

Lagrange Interpolation

Fields of Fractions

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

Modules

Endomorphisms

Representing a Ring

Modules

Submodules

Direct Sums

Free Modules

Vector Spaces

Abelian Groups

Group Actions

Actions

Orbits

Transitive Actions

Fixed Points

Faithful Actions

Cores

Alternating Groups

Sylow Theorems

Quasigroups

Quasigroups

Latin Squares

Division

Quasigroup Homomorphisms

Quasigroup Homotopies

Principal Isotopy

Loops

Index

Exercises, Study Projects, and Notes appear at the end of each chapter.