A Book of Abstract Algebra

Citation

Pinter, C. C. (2010). A Book of Abstract Algebra (2nd ed.). Dover Publications. Unabridged republication of the second edition (McGraw-Hill, 1990); first edition 1982.

Chapter Summary

Each of the 33 short chapters is a discussion of one central idea followed by long exercise sets; topics italicised in the contents appear only in the exercises.

Chapter 1: Why Abstract Algebra?

A historical introduction: the history of algebra, new algebras, algebraic structures, axioms and axiomatic algebra, and abstraction. The preface says the book opens with the promise of showing that the quintic cannot be solved by radicals and keeps it in Chapter 33.

Chapters 2–5: Operations; The Definition of Groups; Elementary Properties of Groups; Subgroups

Operations and their properties lead to groups (finite and infinite, abelian and nonabelian, with group tables), their elementary properties, subgroups, and generators and defining relations. Exercise sets add maximum-likelihood decoding, direct products, Cayley diagrams, the center of a group and group codes.

Chapters 6–8: Functions; Groups of Permutations; Permutations of a Finite Set

Injective, surjective and bijective functions are reviewed just before permutation groups (exercises on finite-state machines and automata). Symmetric and dihedral groups follow, with an exercise application to anthropology, then cycles, transpositions, even and odd permutations, and alternating groups.

Chapters 9–11: Isomorphism; Order of Group Elements; Cyclic Groups

Isomorphic and nonisomorphic groups, Cayley's theorem and (in exercises) automorphisms; then powers of elements, laws of exponents, the order of an element, and cyclic groups with their subgroups.

Chapters 12–13: Partitions and Equivalence Relations; Counting Cosets

Equivalence relations prepare for Lagrange's theorem and its consequences; the exercises survey groups of order at most 10, conjugate elements and group actions.

Chapters 14–16: Homomorphisms; Quotient Groups; The Fundamental Homomorphism Theorem

Homomorphisms, normal subgroups and kernels lead to quotient groups and the fundamental homomorphism theorem. The exercise sets go much further: the class equation, the isomorphism and correspondence theorems, Cauchy's and Sylow's theorems, and the decomposition of finite abelian groups.

Chapters 17–23: Rings: Definitions and Elementary Properties; Ideals and Homomorphisms; Quotient Rings; Integral Domains; The Integers; Factoring into Primes; Elements of Number Theory (Optional)

Rings, integral domains and fields; ideals, quotient rings, prime and maximal ideals; characteristic, finite fields and the field of quotients. The integers are characterised as an ordered integral domain, leading to unique factorisation. Optional Chapter 23 covers congruences, Fermat's and Euler's theorems and the Chinese remainder theorem.

Chapters 24–26: Rings of Polynomials; Factoring Polynomials; Substitution in Polynomials

Polynomial rings, the division algorithm over a field, irreducible polynomials and unique factorisation in F[x], then roots and factors, polynomials over ℚ and the reals, Eisenstein's criterion and interpolation.

Chapters 27–30: Extensions of Fields; Vector Spaces; Degrees of Field Extensions; Ruler and Compass

Algebraic and transcendental elements and the minimum polynomial; a short chapter of linear algebra (independence, basis, dimension, linear transformations) used to define the degree of an extension; simple and iterated extensions; and constructible numbers and impossible constructions.

Chapters 31–33: Galois Theory: Preamble; Galois Theory: The Heart of the Matter; Solving Equations by Radicals

Multiple roots, root fields and isomorphisms of extensions; field automorphisms, the Galois group, the Galois correspondence and the fundamental theorem; and radical extensions and solvable groups, ending with the insolvability of the quintic.

Appendices A–C: Review of Set Theory; Review of the Integers; Review of Mathematical Induction

Short reviews with their own exercises, added in the second edition for courses that start with preliminaries, followed by answers to selected exercises.

Key Concepts

Chapters 1–13: Groups

• Group: a set with an associative operation, an identity element and inverses; abelian when the operation is commutative.
• Cycle decomposition: every permutation of a finite set is a product of disjoint cycles and is either even or odd.
• Isomorphism: a bijection that preserves the operation; Cayley's theorem shows that every group is isomorphic to a group of permutations.
• Lagrange's theorem: the order of a subgroup of a finite group divides the order of the group.

Chapters 14–16: Homomorphisms and quotients

• Normal subgroup and kernel: the kernel of a homomorphism is always a normal subgroup.
• Fundamental homomorphism theorem: the image of a homomorphism is isomorphic to the quotient of the domain by the kernel.

Chapters 17–26: Rings, integers and polynomials

• Integral domain and field: commutative rings with unity and no zero divisors, and those in which every nonzero element is invertible.
• Prime and maximal ideals: ideals whose quotient rings are integral domains and fields respectively.
• Irreducible polynomial: one that does not factor into polynomials of lower degree over the given field.

Chapters 27–33: Fields and Galois theory

• Degree of an extension: the dimension of the larger field as a vector space over the smaller; it explains why some ruler-and-compass constructions are impossible.
• Galois group: the automorphisms of an extension that fix the base field, matched with intermediate fields by the fundamental theorem.
• Solvable group: the group condition that corresponds to solvability by radicals.

Critical Analysis

Strengths

• Motivation before formalism. The preface explains that the book avoids the definition–theorem–proof–corollary sequence and is written as a discussion with the student, tracing concepts to their origins and to familiar intuitions. Proofs are seldom more than a few lines long.
• Background supplied when it is needed. Functions are reviewed just before permutation groups and equivalence relations just before Lagrange's theorem, and Appendices A–C review sets, the integers and induction, so the course does not depend on a separate preliminaries course.
• Exercises are the core of the book. In Allen Stenger's MAA Reviews assessment, over half the space goes to problem sequences. The titled sets move from computation to short proofs to sets that prove a new result, and they reach advanced material such as Sylow's theorem, quadratic residues and algebraic closure, so stronger students have room to go further.
• A clear destination. The polynomial, extension and construction chapters all lead to the insolvability of the quintic, so the later chapters have a visible purpose.

Limitations

• Much theory lives only in the exercises. The isomorphism theorems, the correspondence theorem, Cauchy's and Sylow's theorems and the structure of finite abelian groups appear only in exercise sets. A course that assigns few exercises will miss them.
• Limited help with answers. Only exercises marked # have answers, and the preface describes these as partial solutions or decisive hints.
• Conventional, fixed scope. Stenger describes the choice and treatment of topics as very conventional, and the Dover printing is an unaltered reprint of the 1990 edition. Unique factorisation is treated for ℤ and F[x] rather than for general domains, and modules and other later topics are absent.
• Applications kept to the margins. The second-edition preface states that applications (chiefly coding and automata theory) were added only in exercise sections and may be omitted.

Applications in the Book

Applications appear only in exercise sets: maximum-likelihood decoding (Chapter 3), group codes and the Hamming code (Chapter 5), finite-state machines and automata (Chapter 6), and permutation groups in anthropology (Chapter 7). Cryptography is not among them.

Where It Fits in the Course

Pinter is a discussion-style first course that covers groups, rings and fields and ends with Galois theory. Nathan Carter's Visual Group Theory covers groups only, through pictures and companion software; it works well before or alongside Pinter's Chapters 1–16, and Pinter then supplies the rings, fields and Galois theory that Carter leaves out. Thomas Judson's free Abstract Algebra: Theory and Applications covers similar ground with cryptography and coding theory as full chapters and optional Sage exercises; use it alongside Pinter for applications or computer experiments. Joseph Gallian's Contemporary Abstract Algebra is a larger textbook with many examples and historical notes, suited to students who want more breadth or have already met proofs in a discrete mathematics course.

After Pinter, Peter Cameron's Introduction to Algebra takes rings before groups and adds modules, the construction of the number systems, and further topics; it is a natural next step for the parts Pinter leaves to exercises or omits. R. B. J. T. Allenby's Rings, Fields and Groups is a more rigorous and comprehensive treatment of the same three structures, best read after a first course rather than instead of one.

How to Use This Book

The book suits a first course in abstract algebra. Dover describes it as suitable for junior- and senior-level mathematics majors and future mathematics teachers, and the preface says it is aimed at the average student. No earlier algebra course is assumed: elementary background is filled in as it is needed, and what is required is school algebra and a willingness to read and write short proofs. Review the appendices first if sets, the integers or induction are unfamiliar.

A practical plan:

• First semester (groups): Chapters 1–16. Read Chapter 1 quickly for orientation; Chapters 12–16 are the heart of the group theory and need the most time.
• Second semester or second half (rings and fields): Chapters 17–22 and 24–27, then 29–30. Chapter 23 is marked optional. Chapter 28 can be skimmed by students who have taken linear algebra.
• Galois theory: Chapters 31–33, which depend on Chapters 24–29 and on the quotient-group material of Chapters 14–16.
• One-semester survey: Chapters 1–19, skimming Chapters 2, 6 and 12 if operations, functions and equivalence relations are familiar, then Chapters 24, 25 and 27 as time allows.

Treat the exercise sets as the main text. In each chapter, do the opening computational sets, at least one short-proof set and one later set that proves a new result; follow sets marked † in order. Exercises marked # have partial answers at the back; for the rest, a study partner or instructor is useful for checking proofs. Students heading for further algebra should work the exercise sets on the isomorphism theorems and Sylow's theorem in Chapter 16, since the main text does not prove them. The application exercise sets in Chapters 3 and 5–7 are optional and can be added for interest.