Introduction to Algebra

Citation

Cameron, P. J. (2008). Introduction to Algebra (2nd ed.). Oxford University Press.

Chapter Summary

Chapter 1: Introduction

The opening chapter supplies the background a first-year student needs before abstract algebra: what a proof is, properties of numbers, algebraic manipulation, sets, functions and relations (equivalence relations above all), modular arithmetic, matrices and permutations, with an appendix on logic. It was more than doubled in length for the second edition so that the book could serve a first-year course, and it moves at a deliberately gentle pace.

Chapter 2: Rings

Rings come before groups, by design: everyone knows the integers, which satisfy the ring axioms and give a reliable guide in the early stages. The chapter covers rings and subrings, homomorphisms and ideals, factorisation (leading to a proof of the Fundamental Theorem of Arithmetic) and fields, with an appendix of miscellaneous material.

Chapter 3: Groups

Groups and subgroups, cosets and Lagrange's Theorem, homomorphisms and normal subgroups are developed in parallel with the ring chapter, so that isomorphism theorems and quotients appear for the second time. A section on special groups covers families such as cyclic, symmetric and alternating groups, and an appendix asks how many groups of a given order there are.

Chapter 4: Vector spaces

This short chapter does not try to be a complete account of linear algebra, since most readers will have met vector spaces already. It restates the subject axiomatically (subspaces, bases and dimension, linear transformations and matrices) in a form that generalises to modules in the next chapter.

Chapter 5: Modules

Modules generalise vector spaces by letting the scalars come from a ring. After an introduction, the chapter studies modules over a Euclidean domain, using row and column operations to reach the Smith normal form, and then applies the result to two problems: the structure of finitely generated abelian groups and canonical forms of matrices (with the Cayley–Hamilton Theorem).

Chapter 6: The number systems

The number systems are built from the ground up, from the natural numbers to the complex numbers, so that the earlier assumption that the integers form a ring rests on a proper foundation. The chapter distinguishes algebraic from transcendental numbers, proves that certain ruler-and-compass constructions are impossible, and ends with more set theory: countability, and the Axiom of Choice and Zorn's Lemma, which are new in this edition.

Chapter 7: Further topics

The longest chapter carries Chapters 2 and 3 further. Further group theory includes group actions, Sylow's Theorem and composition series; further ring theory includes unique factorisation domains and local rings; further field theory includes finite fields. A final part introduces structures chosen for their unifying role: universal algebra, lattices and Boolean algebras, and categories. The author describes some of this material as too sketchy to teach from; it is meant to invite further reading.

Chapter 8: Applications

Two applications draw on most of the book. Coding theory covers error-correcting codes, linear codes with generator and check matrices, syndrome decoding, bounds on code size and cyclic codes built from finite fields. Galois theory covers splitting fields, the Galois correspondence and its use to decide which polynomial equations are solvable by radicals.

Key Concepts

Chapter 1: Introduction

• Equivalence relation: a reflexive, symmetric and transitive relation, whose classes partition a set; the author stresses how much algebra depends on it.
• Modular arithmetic: arithmetic on remainders, the first example of a finite algebraic structure.

Chapters 2–3: Rings; Groups

• Ideal and quotient ring: an ideal is the kernel of a ring homomorphism, and the quotient by it is a new ring.
• Factorisation in rings: unique factorisation in suitable rings, which yields the Fundamental Theorem of Arithmetic.
• Normal subgroup and quotient group: the group analogue of an ideal, the kernel of a group homomorphism.
• Lagrange's Theorem: the order of a subgroup divides the order of a finite group.

Chapters 4–5: Vector spaces; Modules

• Module over a ring: a vector space in which the scalars form a ring instead of a field.
• Smith normal form: a diagonal form of a matrix over a Euclidean domain reached by row and column operations, the key tool of Chapter 5.
• Structure of finitely generated abelian groups: such a group is a direct sum of cyclic groups, obtained as an application of the module theory.

Chapter 6: The number systems

• Construction of the number systems: the integers, rationals, reals and complex numbers built in turn on the natural numbers.
• Algebraic and transcendental numbers: numbers that are, or are not, roots of polynomials with rational coefficients; the distinction behind the impossibility of certain constructions.
• Zorn's Lemma: a form of the Axiom of Choice used to prove existence results such as bases of infinite-dimensional spaces.

Chapter 7: Further topics

• Sylow's Theorem: the existence and conjugacy of subgroups of prime-power order in a finite group.
• Finite fields: fields with a prime-power number of elements, whose multiplicative group is cyclic.
• Universal algebra and lattices: general frameworks that treat groups, rings and other structures as instances of one idea.

Chapter 8: Applications

• Linear code and minimum distance: a subspace of words over a finite field whose minimum distance determines how many errors it corrects.
• Galois correspondence: a matching between the subgroups of the Galois group and the intermediate fields of an extension.

Critical Analysis

Strengths

• A clear and argued route into abstraction. Cameron explains why he treats rings before groups: the integers make the axioms feel familiar, the factorisation theorems prove real facts about numbers, and the path to substantial applications is shorter. Students who find group axioms abstract at first may find this order easier.
• A gentle start for first-year students. The expanded Chapter 1 covers proof, sets, relations, matrices and permutations, so a student who has only school mathematics can begin here. zbMATH's reviewer of the second edition singles out this change as the most significant one.
• Unusual breadth for one volume. Besides groups, rings and fields, the book covers modules with their two classical applications, a construction of the number systems, set-theoretic tools such as Zorn's Lemma, introductions to universal algebra, lattices and categories, and two full applications. One book can therefore support several different courses.
• Plenty of exercises and online support. Each chapter has a substantial exercise set, with harder problems marked by an asterisk. The author's website provides solutions to the odd-numbered exercises, extra problems on groups and rings, and a list of misprints.

Limitations

• The pace increases. The preface warns that Chapter 1 is gentle, Chapters 2 and 3 are quicker, and later chapters are faster again. Readers who chose the book for its introduction should expect much less hand-holding from Chapter 4 on.
• Some topics are only sketched. The author himself describes parts of Chapter 7, especially its final survey of other structures, as too sketchy to teach from. Galois theory and coding theory each get a single section of under twenty pages, enough for the main theorems but not for a full course without supplementary reading.
• Linear algebra is assumed. Chapter 4 deliberately omits much of linear algebra, so readers without a matrix and vector-space course will find Chapters 4 and 5 hard going.
• Planned repetition and forward references. Because the book is meant to serve several courses, some ideas first appear in exercises and are treated again later. This helps readers who dip into the book, but can feel uneven to someone reading straight through.

Applications in the Book

The applications are mathematical rather than industrial. Chapter 5 applies modules over Euclidean domains to classify finitely generated abelian groups and to obtain canonical forms of matrices. Chapter 6 uses the theory of algebraic numbers to show that certain ruler-and-compass constructions are impossible. Chapter 8 applies linear algebra over finite fields to error-correcting codes (generator and check matrices, syndrome decoding, bounds such as the Hamming and Singleton bounds, and cyclic codes) and develops Galois theory to explain why there is no general formula for solving polynomial equations of degree five by radicals.

Where It Fits in the Course

Within this course list, the closest comparison is Gallian's Contemporary Abstract Algebra, an undergraduate textbook for a first course with careful proofs, many examples and exercises, and historical notes. Thomas Judson's free Abstract Algebra: Theory and Applications is a similar first-course text on groups, rings and fields, with applications and optional Sage exercises. Nathan Carter's Visual Group Theory builds intuition for groups through pictures before formal definitions, which makes it a good preliminary or companion for Chapter 3. Cameron covers the same core but puts rings before groups and reaches further, into modules, the construction of the number systems, and universal algebra and lattices. Students taking a standard first course who want more examples should pair Cameron with Gallian or Judson. Students who want one book for both a first and a second algebra course will get more out of Cameron.

Pinter's A Book of Abstract Algebra is an accessible first course that develops group theory over a series of short chapters before turning to rings and fields. It can be read before Cameron or alongside Chapters 2 and 3, and its groups-first order is a useful contrast to Cameron's. Allenby's Rings, Fields and Groups is another introduction to abstract algebra. Like Cameron, it treats rings before groups, after opening chapters on numbers, polynomials and relations, and it ends with a brief excursion into Galois theory. Students who like Cameron's rings-first route can use it for a second explanation of the same ideas.

How to Use This Book

The book is written for mathematics undergraduates from the first year onward. The only prerequisite for Chapters 1–3 is school mathematics plus a willingness to read and write proofs. From Chapter 4 on, a first course in linear algebra is assumed.

Choose a route instead of reading straight through. The prefaces suggest several:

• A first algebra course: Chapter 1 as needed, then Chapters 2 and 3.
• A second course: Chapters 5 and 7.
• A group theory course: Chapter 3 with the group theory sections of Chapter 7.
• A ring theory course: Chapters 2 and 5 with the ring theory sections of Chapter 7.
• Galois theory: the field sections of Chapters 2, 6 and 7, then the Galois theory part of Chapter 8.
• Coding theory: the finite-field material in Chapters 2 and 7, then the coding theory part of Chapter 8.

Students who already know proofs and sets can skim Chapter 1, but the sections on relations and modular arithmetic are used throughout. Treat the last part of Chapter 7 as optional further reading.

Do the exercises as you go: some introduce ideas that the text uses later. The asterisked ones are harder. Check your work against the author's online solutions to the odd-numbered exercises and look at the misprints list. The solutions for Chapter 7 are incomplete, so for the further topics a second text or an instructor is useful. The author's solutions also show his notation: maps are written to the right of their arguments and matrices act on row vectors, which takes some getting used to. When the treatment of Galois or coding theory feels brief, follow up with the book's "Further reading" list.