Abstract algebra is a branch of mathematics that studies the properties and structures of mathematical objects such as groups, rings, and fields. “Rings, Fields and Groups: An Introduction to Abstract Algebra” by R. B. J. T. Allenby provides a comprehensive introduction to this area of mathematics.
The book covers the fundamental concepts of abstract algebra, including groups, rings, and fields, and provides an in-depth exploration of each of these topics. The author presents the material in a clear and concise manner, making the book accessible to students with a solid foundation in mathematics. The book also includes numerous examples and exercises that provide opportunities for students to practice and apply their knowledge.
One of the strengths of the book is its focus on the applications of abstract algebra. The author provides numerous examples and applications that demonstrate the practical significance of abstract algebra in various fields such as computer science, engineering, and physics. These applications provide a real-world context that helps to illustrate the abstract concepts discussed in the book.
Numbers and polynomials
- Introduction
- The basic axioms. Mathematical induction
- Divisibility, irreducibles and primes
- Biography and portrait of Hubert
- GCDs
- The unique factorisation theorem (two proofs)
- Polynomials—what are they?
- The basic axioms
- The ‘new’ notation
- Divisibility, irreducibles and primes in Q[x]
- The division algorithm
- Roots and the remainder theorem
Binary relations and binary operations
- Introduction
- Congruence mod n. Binary relations
- Equivalence relations and partitions
- Biography and portrait of Gauss 68
- Some deeper number-theoretic results concerning
- Functions
- Binary operations
Introduction to rings
- Introduction
- The abstract definition of a ring
- Biography and portrait of Hamilton
- Ring properties deducible from the axioms
- Subrings, subfields and ideals
- Biography and portrait of Noether
- Biography and portrait of Fermat
- Fermat’s conjecture (FC)
- Divisibility in rings
- Euclidean rings, unique factorisation domains and principal ideal domains
- Three number-theoretic applications
- Biography and portrait of Dedekind
- Unique factorisation reestablished. Prime and maximal ideals
- Isomorphism. Fields of fractions. Prime subfields
- U[x] where U is a UFD
- Ordered domains. The uniqueness of Z
Factor rings and fields
- Introduction
- Return to roots. Ring homomorphisms. Kronecker’s theorem
- The isomorphism theorems
- Constructions of R from Q and of C from R
- Biography and portrait of Cauchy
- Finite fields
- Biography and portrait of Moore
- Constructions with compass and straightedge
- Symmetric polynomials
- The fundamental theorem of algebra
Basic group theory
- Biography and portrait of Lagrange
- Axioms and examples
- Deductions from the axioms
- The symmetric and the alternating groups
- Subgroups. order of an element
- Cosets of subgroups. Lagrange’s theorem
- Cyclic groups
- Isomorphism. Group tables
- Biography and portrait of Cayley
- Homomorphisms. Normal subgroups
- Factor groups. The first isomorphism theorem
- Space groups and plane symmetry groups
Structure theorems of group theory
- Normaliser. Centraliser. Sylow’s theorems
- Direct products
- Finite abelian groups
- Soluble groups. Composition series
- Some simple groups
A brief excursion into Galois theory
- Biography and portrait of Galois
- Radical Towers and Splitting Fields
- Some Galois groups: their orders and fixed fields
- Separability and Normality
- Subfields and subgroups
- The groups Gal(R/F) and Gal(S1/F)
- The groups
- A Necessary condition for the solubility of a polynomial equation by radicals
- Biography and portrait of Abel
- A Sufficient condition for the solubility of a polynomial equation by radicals
- Non-soluble polynomials: grow your own!
- Galois Theory—old and new
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In addition to its comprehensive coverage of abstract algebra, the book also includes numerous exercises and problems that provide opportunities for students to practice and apply their knowledge. These exercises range from straightforward computations to more challenging proof-based problems, and they serve as a valuable tool for students to test their understanding and deepen their knowledge of the material.
The book is written in a clear and concise style that is accessible to students with a solid foundation in mathematics. The author provides a step-by-step explanation of the concepts and theorems, making the book suitable for self-study or as a textbook for a course on abstract algebra.
In conclusion, “Rings, Fields and Groups: An Introduction to Abstract Algebra” by R. B. J. T. Allenby is an excellent resource for anyone interested in abstract algebra. It provides a comprehensive overview of the subject, and its focus on applications makes it an ideal resource for students and researchers alike.
We have this book in PDF format, ready for download. Whether you are a student, researcher, or mathematician, this book is an essential resource for anyone interested in abstract algebra. Get your copy today!
