Rings, Fields and Groups, An Introduction to Abstract Algebra

Citation

Allenby, R. B. J. T. (1991). Rings, Fields and Groups: An Introduction to Abstract Algebra (2nd ed.). Edward Arnold.

Chapter Summary

The second edition is a compact undergraduate text of about 380 pages, part of Edward Arnold's Modular Mathematics series. Rings and fields come before groups, so the book moves from the familiar arithmetic of integers and polynomials through ring theory to group theory, and closes with a chapter on Galois theory. The publisher says the author mixes informal and formal material to stimulate the student's enthusiasm while still giving the theoretical concepts needed for serious study. The book also has partial solutions, a bibliography, a list of notation and an index at the end.

Elementary Set Theory and Methods of Proof

An opening chapter before the numbered chapters collects the set theory and proof techniques used throughout, so a reader new to proofs has the groundwork in the book itself.

Chapter 1: Numbers and Polynomials

The chapter studies the integers and polynomials side by side: divisibility, primes and the fundamental theorem of arithmetic, then polynomial rings, divisibility in them and their algebraic properties. The Euclidean algorithm for integers and polynomials sets the pattern for later chapters.

Chapter 2: Binary Relations and Binary Operations

Binary relations and operations are introduced, with congruences modulo n, equivalence relations and partitions. These tools are the basis for the quotient constructions later in the book.

Chapter 3: Introduction to Rings

The abstract definition of a ring, properties deduced from the axioms, subrings, subfields and ideals. Number-theoretic applications of ring theory are included.

Chapter 4: Factor Rings and Fields

Factor rings, ring homomorphisms and the isomorphism theorems are developed, and fields are constructed from rings by passing to a quotient, with constructions linking different fields.

Chapter 5: Basic Group Theory

Groups are introduced through the axioms and examples, then subgroups, homomorphisms, normal subgroups and factor groups, and the symmetric and alternating groups. Lagrange's theorem on the order of a subgroup is part of this material.

Chapter 6: Structure Theorems of Group Theory

Further group theory: Sylow's theorems, direct products, finite abelian groups, soluble groups and composition series.

Chapter 7: A Brief Excursion into Galois Theory

The second edition added this chapter. It introduces radical towers, splitting fields and the correspondence between subfields and subgroups, with the insolvability of the general quintic by radicals as its principal application.

Key Concepts

Chapters 0–2: Foundations

• Divisibility, primes and polynomial rings: arithmetic of integers and polynomials.
• Equivalence relations and congruences: how relations partition a set and give modular arithmetic.

Chapters 3–4: Rings and fields

• Rings, subrings and ideals: the basic objects of ring theory.
• Factor rings, homomorphisms and isomorphism theorems: constructing new rings from old and comparing them.
• Field construction: obtaining fields from rings by taking quotients.

Chapters 5–7: Groups and Galois theory

• Groups, subgroups, normal subgroups and factor groups.
• Sylow's theorems and the structure of finite abelian groups.
• Splitting fields and the Galois correspondence.

Critical Analysis

Strengths

• A deliberate order. Rings and fields precede groups, so students build on arithmetic they already know, and the quotient construction is met first in a familiar setting before it reappears for groups.
• Exercises with solutions. The publisher notes about 800 exercises, with hints and solutions to many of them, and the book ends with partial solutions, which helps self-study.
• Readable style. Mathematical Spectrum is quoted by the publisher as calling it no dry-as-dust textbook, one that pays dividends to readers who give it time and effort. The publisher describes the style as highly readable.
• Complete arc. The book takes a reader from elementary arithmetic to a first account of Galois theory in a single volume.

Limitations

• Galois theory is only an excursion. The final chapter is titled a brief excursion, so a reader who wants the subject developed in full will need another text.
• Scope. There are seven numbered chapters, with no separate treatment of topics such as modules or linear algebra over rings, and the book keeps to rings, fields, groups and an introduction to Galois theory.

Where It Fits in the Course

This is an undergraduate first course in abstract algebra that takes rings before groups. A Book of Abstract Algebra is a conversational first course that also ends with Galois theory and relies heavily on exercise sets; Contemporary Abstract Algebra and Abstract Algebra: Theory and Applications are larger undergraduate texts with many examples and applications; Introduction to Algebra is a broad introduction that also takes in vector spaces and modules. Visual Group Theory is a picture-based route into groups, useful alongside Chapter 5.

Allenby suits readers who want a compact text with partial solutions at the back and a short route to Galois theory. Readers who prefer a conversation-style introduction may begin with A Book of Abstract Algebra, and those who want a fuller treatment of group theory can pair Chapters 5–6 with Contemporary Abstract Algebra.

How to Use This Book

The publisher gives the readership as undergraduate students of pure mathematics and describes the book as an introduction to the material studied in undergraduate abstract algebra courses. It needs mathematical maturity and some comfort with proofs, but the opening chapter on set theory and methods of proof means a first proof course is not a strict prerequisite.

A practical plan:

• Core: the opening chapter as needed, then Chapters 1–5 in order. Chapters 3 and 4 on rings and quotients are the heart of the first half; Chapter 5 starts the group theory.
• Next: Chapter 6 for Sylow's theorems and the structure of finite abelian groups.
• Optional capstone: Chapter 7 on Galois theory, best read after Chapters 4 and 6.
• Exercises: attempt each exercise before consulting the hints or the partial solutions at the end of the book, and use the solutions to check technique rather than to shortcut the work.