Elements of Modern Algebra” by Gilbert, 7th Edition is a widely-used textbook that provides students with a comprehensive introduction to modern algebra. The book is designed to help students understand the fundamental concepts and methods of algebra, and to develop their skills in solving abstract mathematical problems.
One of the strengths of “Elements of Modern Algebra” is its focus on helping students develop a deep understanding of the subject. The author uses clear explanations, numerous examples, and well-chosen exercises to help students see how the concepts fit together and how they can be applied in real-world situations. Throughout the book, the author emphasizes the importance of mathematical reasoning and critical thinking, helping students to develop these essential skills.
The 7th Edition of “Elements of Modern Algebra” covers a wide range of topics, including group theory, ring theory, and field theory. The author provides a thorough introduction to each topic, and then builds on this foundation to explore more advanced concepts. This approach helps students to develop a deep and lasting understanding of the subject, and to see the connections between different areas of algebra.
TABLE OF CONTENTS
Fundamentals
- Sets
- Mappings
- Properties of Composite Mappings
- Binary Operations
- Permutations and Inverses
- Matrices
- Relations
The Integers
- Postulates for the Integers
- Mathematical Induction
- Divisibility
- Prime Factors and Greatest Common Divisor
- Congruence of Integers
- Introduction to Coding Theory
- Introduction to Cryptography
Groups
- Definition of a Group
- Properties of Group Elements
- Subgroups
- Cyclic Groups
- Isomorphisms
- Homomorphisms
.
More on Groups
- Finite Permutation Groups
- Cayley’s Theorem
- Permutation Groups in Science and Arts
- Cosets of a Subgroup
- Normal Subgroups
- Quotient Groups
- Direct Sums
- Some Results on Finite Abelian Groups
Rings, Integral Domains, and Fields
- Definition of a Ring
- Integral Domains and Fields
- The Field of Quotients of an Integral Domain
- Ordered Integra Domains
More on Rings
- Ideals and Quotient Rings
- Ring Homomorphisms
- The characteristic of a Ring
- Maximal Ideals
Real and Complex Numbers
- The Field of Real Numbers
- Complex Numbers and Quaternions
- De Moivre’s Theorem and Roots of Complex Numbers
Polynomials
- Ploynomials over a Ring
- Divisibility and Greatest Common Divisor
- Factorization in F[x]
- Zeros of a Polynomial
- Solution of Cubic and Quadratic Equations by Formulas
- Algebraic Extensions on Field
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Another strength of the book is its focus on applications. The author includes numerous examples and exercises that show students how the concepts they are learning can be used in a variety of practical settings. This approach helps students to see the relevance of algebra to their daily lives, and to develop a greater appreciation for the subject.
In conclusion, “Elements of Modern Algebra” by Gilbert, 7th Edition is an excellent textbook for students who are studying modern algebra for the first time, or for those who want to deepen their understanding of the subject. The book is well-written, clear, and comprehensive, and provides students with the knowledge and skills they need to succeed in modern algebra.
For the convenience of students, this book is available in PDF format for download, making it easier for students to access the information they need and study on-the-go.
