Basic Abstract Algebra, written by Robert B. Ash, is a comprehensive textbook designed to help students understand the fundamental concepts of abstract algebra. This book covers the essential topics of group theory, ring theory, and field theory in a clear and concise manner, making it an ideal resource for students who are new to abstract algebra or are looking to build a strong foundation in this area of mathematics.
One of the key features of the Basic Abstract Algebra book is its straightforward and accessible writing style. The author has done an excellent job of breaking down complex mathematical concepts into easy-to-understand terms and explanations, making the book ideal for students of all levels. Additionally, the book is filled with a variety of examples and practice problems that help students reinforce what they have learned and gain a deeper understanding of the material.
TABLE OF CONTENTS
Chapter 0 Prerequisites
- Elementary Number Theory
- Set Theory
- Linear Algebra
Chapter 1 Group Fundamentals
- Groups and Subgroups
- Permutation Groups
- Cosets, Normal Subgroups and Homomorphisms
- The Isomorphism Theorems
- Direct Products
Chapter 2 Ring Fundamentals
- Badic Definitions and Properties
- Ideals, Homomorphisms and Quotient Rings
- The Isomorphism Theorems for Rings
- Maximal and Prime Ideals
- Polynomial Rings
- Polynomial Rings
- Unique Factorization
- Principal Ideal Domains and Euclidean Domains
- Rings of Fractions
- Irreducible Polynomials
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Chapter 3 Field Fundamentals
- Field Extensions
- Splitting Fields
- Algebraic Closures
- Separability
- Normal Extensions
Chapter 4 Module Fundamentals
- Modules and Algebras
- The Isomorphism Theorems for Modules
- Direct Sums and Free Modules
- Homomrphisms and Matrices
- Smith Normal Form
- Fundamental Structure Theorems
- Exact Sequences and Diagram Chasing
Chapter 5 Some Basic Techniques of Group Theory
- Groups Acting on Sets
- The Orbit-Stabilizer Theorem
- Applications to Combinatorics
- The Sylow Theorems
- Applications of the Sylow Theorems
- Applications of the Sylow Theorems
- Composition Series
- Solvable and Nilpotent Groups
Chapter 6 Galois Theory
- Fixed Field and Galois Groups
- The Fundamental Theorem
- Computing a Galois Group Directly
- Finite Fields
- Cyclotomic Fields
- The Galois Group of a Cubic
- Cyclic and Kummer Extensions
- Solvability by Radicals
- Transcendental Extensions
Chapter 7 Introducing Algenraic Number Theory
- Integral Extensions
- Quadratic Extensions os the Rationals
- Norms and Traces
- The Discriminant
- Noetherian and Artinian Modules and Rings
- Fractional Ideals
- Unique Factorization of Ideals in a Dedekind Domain
- Some Arithematic in Dedekind Domains
- P-adic Numbers
Chapter 8 Introducing Algebraic Geometry
- Varieties
- The Hilbert Basis Theorem
- The Nullstellensaz: Preliminaries
- The Nullstellensatz: Equivalnet Versions and Prrof
- Localization
- Primary Decomposition
- Tensor Product of Modules over a Commutative Ring
- General Tensor Products
Chapter 9 Introducing Noncommutative Algebra
- Semisimple Modules
- Two Key Theorems
- Simple and Semisimple Rings
- Further Properties of simple rings, Matrix rings, and Endomorphisms
- The structure of Semisimple Rings
- Maschke’s Theorem
- The Jacobson Radical
- Theorems of Hopkins-Levitzki and Nakayama
Chapter 10 Introducing Homological Algebra
- Categories
- Products and Coproducts
- Functors
- Exact Functors
- Projective Modules
- Injective Modules
- Embedding into an Injective Module
- Flat Modules
- Direct and Inverse Limits
Download Now Book in PDF
Another important aspect of the Basic Abstract Algebra book is its comprehensive approach to teaching abstract algebra. The book covers all of the fundamental concepts that students will need to know in order to succeed in advanced mathematics and beyond. Whether a student is just starting to learn abstract algebra or has been studying it for a while, the Basic Abstract Algebra book has something to offer.
The Basic Abstract Algebra book is also very user-friendly. It has been designed with students in mind and features a well-organized layout that makes it easy to find the information that you need. The book is also filled with helpful tips and tricks that make solving mathematical problems easier and more intuitive.
For students who are looking for additional resources to help them learn and understand abstract algebra, the Basic Abstract Algebra book is available in PDF format for free download. This means that students can access the book from anywhere, at any time, and on any device. Whether they are studying at home, on the go, or in the classroom, they can have the Basic Abstract Algebra book right at their fingertips.
In conclusion, Basic Abstract Algebra by Robert B. Ash is an excellent resource for students who are looking to master the fundamental concepts of abstract algebra. With its straightforward and accessible writing style, comprehensive approach to teaching, and user-friendly design, the Basic Abstract Algebra book is the perfect tool for students who want to excel in mathematics. So, if you are looking for a reliable and effective guide to help you learn abstract algebra, be sure to check out the Basic Abstract Algebra book today!
