Algebra Abstract and Concrete is a comprehensive textbook written by Frederick M. Goodman and published in 2002. The book provides a thorough introduction to abstract algebra, including topics such as group theory, ring theory, and field theory. It also includes concrete examples and applications to help students understand abstract concepts and apply them to real-world problems.
One of the key strengths of this book is its clear and concise writing style. Frederick M. Goodman does an excellent job of explaining abstract concepts in a way that is accessible to students, making it an ideal choice for those who are new to the subject. The book covers a wide range of topics and is well organized, making it easy for students to follow along and understand the material.
In addition to its clear writing style, Algebra Abstract and Concrete is also notable for its comprehensive coverage of the subject matter. The book covers all of the core topics in abstract algebra, including group theory, ring theory, and field theory, and provides a wealth of concrete examples and applications to help students understand and apply the concepts.
TABLE OF CONTENTS
Chapter 1. Algebraic Themes
- What Is Symmetry?
- Symmetries of the Rectangle and the Square
- Multiplication Tables
- Symmetries and Matrices
- Permutations
- Divisibility in the Integers
- Modular Arithmetic
- Polynomials
- Counting
- Groups
- Rings and Fields
- An Application to Cryptography
.
Chapter 2. Basic Theory of Groups
- First Results
- Subgroups and Cyclic Groups
- The Dihedral Groups
- Homomorphisms and Isomorphisms
- Cosets and Lagrange’s Theorem
- Equivalence Relations and Set Partitions
- Quotient Groups and Homomorphism Theorems
Chapter 3. Products of Groups
- Direct Products
- Semidirect Products
- Vector Spaces
- The dual of a vector space and matrices
- Linear algebra over Z
- Finitely generated abelian groups
Chapter 4. Symmetries of Polyhedra
- Rotations of Regular Polyhedra
- Rotations of the Dodecahedron and Icosahedron
- What about Reflections?
- Linear Isometries
- The Full Symmetry Group and Chirality
Chapter 5. Actions of Groups
- Group Actions on Sets
- Group Actions—Counting Orbits
- Symmetries of Groups
- Group Actions and Group Structure
- Application: Transitive Subgroups of S5
- Additional Exercises for Chapter 5
Chapter 6. Rings
- A Recollection of Rings
- Homomorphisms and Ideals
- Quotient Rings
- Integral Domains
- Euclidean Domains, Principal Ideal
- Domains, and Unique Factorization
- Unique Factorization Domains
- Noetherian Rings
- Irreducibility Criteria
Chapter 7. Field Extensions – First Look
- A Brief History
- Solving the Cubic Equation
- Adjoining Algebraic Elements to a Field
- Splitting Field of a Cubic Polynomial
- Splitting Fields of Polynomials in CŒx
Chapter 8. Modules
- The idea of a module
- Homomorphisms and quotient modules
- Multilinear maps and determinants
- Finitely generated Modules over a PID, part I
- Finitely generated Modules over a PID, part II.
- Rational canonical form
- Jordan Canonical Form
Chapter 9. Field Extensions – Second Look
- Finite and Algebraic Extensions
- Splitting Fields
- The Derivative and Multiple Roots
- Splitting Fields and Automorphisms
- The Galois Correspondence
- Symmetric Functions
- The General Equation of Degree n
- Quartic Polynomials
- Galois Groups of Higher Degree Polynomials
Chapter 10. Solvability
- Composition Series and Solvable Groups
- Commutators and Solvability
- Simplicity of the Alternating Groups
- Cyclotomic Polynomials
- The Equation
- Solvability by Radicals
- Radical Extensions
Chapter 11. Isometry Groups
- More on Isometries of Euclidean Space
- Euler’s Theorem
- Finite Rotation Groups
- Crystals
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The book is also designed to be flexible and can be used in a variety of different courses and settings. Whether you are studying algebra in a traditional classroom setting or working through the material on your own, Algebra Abstract and Concrete is an excellent resource that will provide you with a solid foundation in abstract algebra.
For those looking to further their understanding of abstract algebra, Algebra Abstract and Concrete is an excellent choice. Whether you are a student, a teacher, or just someone who is interested in the subject, this book provides a comprehensive and accessible introduction to abstract algebra.
And the good news is, we have this book in PDF format to download, so you can access it anytime and anywhere. The PDF format allows you to easily search the book, highlight key passages, and take notes, making it a convenient and accessible resource for students of all levels.
In conclusion, Algebra Abstract and Concrete is a comprehensive and accessible textbook that is ideal for students and teachers of abstract algebra. Its clear writing style, comprehensive coverage of the subject matter, and flexible design make it a valuable resource for anyone looking to deepen their understanding of abstract algebra.
