Abstract Algebra: Theory and Applications

Citation

Judson, T. W. (2026). Abstract Algebra: Theory and Applications (2026 annual edition; Sage exercises by R. A. Beezer). Orthogonal Publishing L3C. Free official edition: https://judsonbooks.org/abstract-algebra-theory-and-applications/

Chapter Summary

Chapters 1–2: Preliminaries; The Integers

Chapter 1 sets out the background a reader needs: a short note on what proofs are and how to write them, followed by sets, functions, relations and equivalence relations with partitions. Chapter 2 develops the integers as the first algebraic model—mathematical induction and the well-ordering principle, the division algorithm, greatest common divisors via the Euclidean algorithm, and the Fundamental Theorem of Arithmetic.

Chapter 3: Groups

Groups are motivated through integer equivalence classes (ℤₙ) and the symmetries of a triangle, then defined axiomatically with many examples, including matrix groups. Basic properties and subgroups follow. An additional exercise section shows how check digits in identification numbers such as UPC and ISBN codes detect errors.

Chapter 4: Cyclic Groups

The chapter studies groups generated by a single element and proves that subgroups of cyclic groups are cyclic. It treats the multiplicative group of complex numbers, the circle group and roots of unity, and the method of repeated squares for computing large powers modulo n, which prepares for cryptography.

Chapter 5: Permutation Groups

Permutations are written in cycle notation and products of transpositions; even and odd permutations lead to the alternating groups. Dihedral groups are introduced as symmetry groups of regular polygons, along with the motion group of a cube, giving many nonabelian examples.

Chapter 6: Cosets and Lagrange's Theorem

Cosets partition a group, which gives Lagrange's Theorem: the order of a subgroup divides the order of the group. The chapter notes that the converse fails and derives Fermat's and Euler's theorems as consequences.

Chapter 7: Introduction to Cryptography

An application chapter covering private-key systems (shift, affine and matrix ciphers) and public-key cryptography based on Euler's theorem, specifically RSA, including how RSA can be used to verify that a message is authentic. Additional exercises explore primality testing and factoring.

Chapter 8: Algebraic Coding Theory

Error-detecting and error-correcting codes are introduced with Hamming distance; linear codes are treated as subgroups of ℤ₂ⁿ, described by parity-check and generator matrices, and decoded efficiently with syndromes and coset decoding.

Chapter 9: Isomorphisms

Isomorphic groups are defined and studied; Cayley's Theorem shows that every group is isomorphic to a group of permutations. External and internal direct products are introduced, along with conditions for ℤₘ × ℤₙ to be cyclic.

Chapter 10: Normal Subgroups and Factor Groups

Normal subgroups are exactly those for which left and right cosets agree, and they allow the construction of factor groups. The chapter proves that the alternating group Aₙ is simple for n ≥ 5.

Chapter 11: Homomorphisms

Group homomorphisms and kernels generalise isomorphisms. The First, Second and Third Isomorphism Theorems and the Correspondence Theorem are proved, with additional exercises on automorphisms.

Chapter 12: Matrix Groups and Symmetry

Following Klein's view of geometry as the study of invariants, the chapter treats general and special linear groups, orthogonal groups and the Euclidean group, then classifies finite symmetry groups in the plane and describes wallpaper and crystallographic groups.

Chapter 13: The Structure of Groups

The Fundamental Theorem of Finite Abelian Groups classifies finite abelian groups as direct products of cyclic groups of prime-power order. The second half introduces subnormal and composition series, the Jordan–Hölder Theorem and solvable groups, which are needed for Galois theory.

Chapter 14: Group Actions

Groups acting on sets give orbits and stabilisers; the class equation follows from the action of a group on itself by conjugation. Burnside's Counting Theorem is applied to counting colourings and to classifying switching functions.

Chapter 15: The Sylow Theorems

Using group actions, the chapter proves the three Sylow Theorems as a partial converse to Lagrange's Theorem and applies them to show that certain groups are not simple and to classify small groups. It closes with a project on finding groups of small order.

Chapter 16: Rings

Rings, integral domains and fields are defined with examples including matrices, quaternions and polynomials. Ring homomorphisms, ideals, factor rings, and maximal and prime ideals follow, and a final section applies the Chinese Remainder Theorem to designing software for parallel computation with large integers.

Chapter 17: Polynomials

Polynomial rings are studied through the division algorithm and the roots of polynomials. The chapter then covers irreducible polynomials, including Gauss's Lemma and Eisenstein's Criterion, shows that every ideal of F[x] is principal when F is a field, and offers additional exercises on solving cubic and quartic equations.

Chapter 18: Integral Domains

The field of fractions of an integral domain is constructed. Unique factorisation domains, principal ideal domains and Euclidean domains are compared, with the Gaussian integers as a central example and a proof that polynomial rings over UFDs are UFDs.

Chapter 19: Lattices and Boolean Algebras

Partially ordered sets, lattices and Boolean algebras are introduced as structures that generalise set inclusion, union and intersection. Finite Boolean algebras are classified, and the theory is applied to switching circuits.

Chapter 20: Vector Spaces

A brief chapter on vector spaces over arbitrary fields—subspaces, linear independence, bases and dimension—provides the tools needed to treat field extensions.

Chapter 21: Fields

Algebraic and transcendental extensions, the degree of an extension and its multiplicativity, algebraic closure and splitting fields are developed. The theory is applied to classical geometric constructions, showing that it is impossible to double a cube, trisect an arbitrary angle or square a circle with straightedge and compass.

Chapter 22: Finite Fields

The chapter proves that for each prime power there is a unique finite field of that order and that its multiplicative group is cyclic. It applies finite fields to polynomial (cyclic) codes, with additional exercises on BCH codes.

Chapter 23: Galois Theory

Field automorphisms and the Galois group of a polynomial lead to the Fundamental Theorem of Galois Theory, which matches intermediate fields with subgroups. Applications include solvability by radicals, the insolvability of the general quintic, and a proof of the Fundamental Theorem of Algebra.

Key Concepts

Chapters 1–2: Preliminaries; The Integers

• Equivalence relation and partition: a reflexive, symmetric, transitive relation, whose classes split a set into disjoint pieces.
• Division algorithm and Euclidean algorithm: dividing with remainder, and the repeated division that computes a greatest common divisor.

Chapters 3–4: Groups; Cyclic Groups

• Group: a set with an associative operation, an identity and inverses.
• Subgroup: a subset that is itself a group under the same operation.
• Cyclic group: a group generated by one element; every such group is isomorphic to ℤ or some ℤₙ.

Chapters 5–6: Permutation Groups; Cosets and Lagrange's Theorem

• Cycle notation and parity: writing permutations as cycles and classifying them as even or odd by transpositions.
• Dihedral group: the symmetry group of a regular n-gon, with 2n elements.
• Lagrange's Theorem: the order of a subgroup of a finite group divides the order of the group.

Chapters 7–8: Introduction to Cryptography; Algebraic Coding Theory

• RSA: a public-key system whose security rests on the difficulty of factoring a product of two large primes.
• Linear code: a code that is a subgroup of ℤ₂ⁿ, described by generator and parity-check matrices.
• Minimum distance: the smallest Hamming distance between codewords, which determines how many errors can be detected or corrected.

Chapters 9–11: Isomorphisms; Normal Subgroups and Factor Groups; Homomorphisms

• Cayley's Theorem: every group is isomorphic to a group of permutations.
• Factor group: the group of cosets of a normal subgroup.
• First Isomorphism Theorem: the image of a homomorphism is isomorphic to the domain modulo the kernel.

Chapters 12–13: Matrix Groups and Symmetry; The Structure of Groups

• Orthogonal group: the matrices that preserve lengths and angles, used to describe symmetries.
• Fundamental Theorem of Finite Abelian Groups: each finite abelian group is a direct product of cyclic groups of prime-power order.
• Solvable group: a group with a subnormal series whose factors are abelian.

Chapters 14–15: Group Actions; The Sylow Theorems

• Orbit and stabiliser: the set of images of a point under the group, and the subgroup fixing it.
• Burnside's Counting Theorem: the number of orbits equals the average number of fixed points.
• Sylow subgroup: a subgroup of maximal prime-power order, whose existence and number the Sylow Theorems control.

Chapters 16–18: Rings; Polynomials; Integral Domains

• Ideal and factor ring: a subring closed under multiplication by ring elements, used to form quotient rings.
• Maximal ideal: an ideal whose factor ring is a field.
• Irreducible polynomial: a polynomial that cannot be factored into lower-degree polynomials over the given field.
• Unique factorisation domain: an integral domain in which factorisation into irreducibles is unique up to units and order.

Chapter 19: Lattices and Boolean Algebras

• Lattice: a partially ordered set in which every pair has a least upper bound and a greatest lower bound.
• Boolean algebra: a complemented distributive lattice, modelling set operations and switching circuits.

Chapters 20–21: Vector Spaces; Fields

• Extension degree: the dimension of a larger field as a vector space over a smaller one.
• Splitting field: the smallest extension in which a polynomial factors into linear factors.
• Constructible number: a length obtainable by straightedge and compass, which must lie in an extension of degree a power of 2.

Chapters 22–23: Finite Fields; Galois Theory

• Galois field: the unique finite field of order pⁿ, whose nonzero elements form a cyclic group.
• Galois group: the group of automorphisms of a field extension that fix the base field.
• Fundamental Theorem of Galois Theory: a correspondence between intermediate fields and subgroups of the Galois group.

Critical Analysis

Strengths

• Applications integrated with theory. The preface explains that applied examples help motivate students meeting rigorous proof for the first time. Cryptography, coding theory, symmetry, counting, software design, switching circuits, geometric constructions and polynomial codes appear as full chapters or sections, so students see why groups, rings and fields matter.
• Flexible structure with clear guidance. The preface gives a chapter dependency chart and suggested syllabi for a one-semester course, a theory-focused two-semester course, and an application-focused course; applied chapters can be dropped without breaking the theory.
• Support for active reading. Each chapter has reading questions to answer before class, an exercise set ranging from computational to theoretical, and often programming exercises, projects and references. Hints and answers to selected exercises are in an appendix.
• Open and computational. The text is under the GNU Free Documentation Licence, is written in PreTeXt with web, PDF and print versions, and companion Sage exercises (tested with a recent SageMath release) are available in the web and Sage editions.

Limitations

• Breadth over depth in places. Some chapters are short—Chapter 20 on vector spaces is only a few pages and Chapter 19 sits somewhat apart from the main line—so the coverage of module theory, representation theory and commutative algebra beyond the basics is absent.
• Solutions are partial. The appendix gives hints and answers to selected exercises only, often with proofs only sketched, so self-learners will need other ways to check their arguments.
• The PDF lacks the Sage material. The standard PDF omits the Sage sections, which appear in the web version and a separate Sage PDF; readers who want the computational component need those versions.
• Assumes some maturity. Although no formal prerequisites are listed, the preface assumes familiarity with matrices and determinants, and Chapter 1's treatment of proof is brief; students with no prior proof course may need support.

Applications in the Book

Applications occupy whole chapters and sections: error detection in UPC and ISBN check digits (Section 3.6), fast modular exponentiation (Section 4.3), private- and public-key cryptography including RSA (Chapter 7), algebraic coding theory with linear codes and syndrome decoding (Chapter 8), symmetry, wallpaper and crystallographic groups (Chapter 12), Burnside counting of colourings and switching functions (Chapter 14), the Chinese Remainder Theorem in parallel-processing software design (Section 16.5), Boolean algebras and electrical circuits (Chapter 19), straightedge-and-compass constructions (Section 21.3), polynomial and BCH codes (Chapter 22), and solvability of polynomial equations by radicals (Section 23.3).

Where It Fits in the Course

Judson's book and Nathan Carter's Visual Group Theory are the gentlest entries in this Abstract Algebra list. Carter's book is generally described as a visual, intuitive introduction to groups; it suits readers who want pictures of group structure before or alongside Judson's Chapters 3–15. Gallian's Contemporary Abstract Algebra is generally described as a widely used undergraduate text with many applications and exercises, and it covers similar ground to Judson; either can serve as the main text with the other as a source of extra examples.

Pinter's A Book of Abstract Algebra is another first course, written conversationally in 33 short chapters that move from groups to rings, field extensions and Galois theory, with much of the theory placed in its exercise sets; it works well alongside Judson as a second explanation. Cameron's Introduction to Algebra and Allenby's Rings, Fields and Groups both take rings before groups. Cameron goes on to vector spaces, modules, the construction of the number systems, coding theory and Galois theory, so it suits students who want more on rings, fields and Galois theory after Judson's Chapters 16–23.

How to Use This Book

The book is meant for a one- or two-semester undergraduate course, typically for mathematics majors and for science, engineering and computer science students minoring in mathematics. Readers should be comfortable reading proofs and should know basic matrices and determinants; Chapters 1–2 review sets, equivalence relations, induction and divisibility.

For a single semester, the preface suggests Chapters 1–6, 9, 10, 11, 13 (first part), 16, 17, 18 (first part), 20 and 21, adding application chapters according to interest. For an applied emphasis, include Chapters 7, 8, 12, 14, 19 and 22; for a theoretical two-semester course, continue through the Sylow Theorems, finite fields and Galois theory. Answer each chapter's reading questions before working the exercises, check selected answers in Appendix B, and use the Sage exercises in the web version to experiment with groups, rings and fields.

Edition and Sources

This guide is based on the 2026 annual edition of Abstract Algebra: Theory and Applications (title page dated 4 August 2026, © 1997–2026 Thomas W. Judson and Robert A. Beezer, GNU Free Documentation Licence), read in the free PDF linked from the official book page, https://judsonbooks.org/abstract-algebra-theory-and-applications/. It draws on the preface (course plans, dependency chart, exercises, reading questions and Sage), the acknowledgements, the full table of contents including all 23 chapters and appendices, every chapter introduction and section heading, and key statements and applied sections (including Sections 3.6, 4.3, 16.5, 21.3 and 23.3).