Abstract Algebra: Theory and Applications (Judson)

thomas-judson‘s undergraduate algebra text (stephen-f-austin-state-university), first published 1997 and maintained since — the HTML edition was still being updated in 2025. GNU Free Documentation License, “Version 1.2 or any later version… with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts,” which puts it on the same free footing as hefferon-linear-algebra and corral-vector-calculus. 23 chapters, ~400 pages before the solutions and licence appendices.

This closes growth edge 8 — the open-licensed algebra text — opened the same day lal-algebra-1 closed the subject and left the licence half unmet. Pulled deliberately rather than routed in.

Read from the August 2012 PDF mirrored on a Colorado State course page; the current edition lives at abstract.ups.edu, which was unreachable from this host. Chapter structure and definitions below are from the copy actually read.

The structure, and where it differs from Lal

Judson’s order is the standard one and he says so: “groups, then rings, and finally fields.”

• 1–2 Preliminaries, The Integers — proofs, sets, equivalence relations, induction, the division algorithm.
• 3–6 Groups — definitions, subgroups, cyclic groups, permutation and dihedral groups, cosets and Lagrange’s theorem, Fermat and Euler.
• 7–8 Cryptography, Algebraic Coding Theory — public-key crypto, error-correcting and linear codes, parity-check matrices, decoding.
• 9–15 — isomorphisms, normal subgroups and factor groups, homomorphisms and the isomorphism theorems, matrix groups and symmetry, structure of groups (finite abelian, solvable), group actions and the class equation, the Sylow theorems.
• 16–18 Rings — rings, integral domains and fields, ideals (maximal and prime), polynomial rings, irreducibility, fields of fractions, factorisation.
• 19–20 — lattices and Boolean algebras (with the algebra of electrical circuits), vector spaces.
• 21–23 Fields — extension and splitting fields, geometric constructions, finite fields, and Galois theory through the fundamental theorem.

Against lal-algebra-1, which is the same subject with the opposite emphasis, three differences stand out. Lal opens with two chapters of logic and set theory and Judson with a short note on proofs. Judson spends two full chapters on applications (cryptography, coding theory) a third of the way in, and returns to applications repeatedly; Lal’s applied content is arithmetic itself. And Judson reaches Galois theory in one volume where Lal defers it to a later one, at the cost of the structure theory Lal spends two chapters on (Remak–Krull–Schmidt, Jordan–Hölder, nilpotent groups, presentations).

Why applications, in the author’s own words

“one of the major problems in teaching an abstract algebra course is that for many students it is their first encounter with an environment that requires them to do rigorous proofs. Such students often find it hard to see the use of learning to prove theorems and propositions; applied examples help the instructor provide motivation.”

That is a pedagogical argument for content selection, and it belongs to the same family as the arguments this corpus has been collecting: Keisler changing the foundation to buy intuition, Stirzaker choosing formality to avoid student error, Corral rating his own rigour at 5/10. Judson keeps full rigour and changes the motivation instead — coding theory as the reason to care about a quotient ring. Unmeasured, like all of them.

Prerequisites and tooling

“There are no specific prerequisites for a course in abstract algebra,” with the caveat that students who have had higher-level mathematics arrive with more maturity, and an occasional assumption of basic linear algebra. A chapter dependency chart in the preface tells an instructor what may be skipped — chapters 1–6 are the trunk.

Rob Beezer contributed Sage material, and the Runestone HTML edition runs the Sage cells in the browser. That puts it beside hefferon-linear-algebra‘s Sage lab manual on the spoke’s computational-tool thread: two of the corpus’s free texts name the same open CAS, where corral-vector-calculus assumes none.

Connections

• author thomas-judson · stephen-f-austin-state-university
• the free counterpart to lal-algebra-1; together they are abstract-algebra‘s two treatments
• group-theory is the bulk of both books
• the field chapters are the structure under linear-algebra‘s scalars