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The chapters Groups, Cyclic Groups, Permutation Groups, Cosets and Lagrange’s Theorem, Introduction to Cryptography, Algebraic Coding Theory, Isomorphisms, Normal Subgroups and Factor Groups, Homomorphisms, Matrix Groups and Symmetry, The Structure of Groups, and Lattices and Boolean Algebras were removed.

Two new chapters, Groups and The Simplicity of \(A_n\text{,}\) were written to reintroduce, in a much shorter form, the specific definitions, theorems, and proofs from the removed chapters that this edition’s remaining content actually depends on — without restoring those chapters in full.

Nicholas Touikan revised several proofs by hand to remove dependencies that the new chapter order had made circular. In the Groups chapter, he added a proof that a finite abelian group has an element whose order equals the group’s exponent, and used it to give a self-contained proof that the multiplicative group of a finite field is cyclic, no longer relying on the Fundamental Theorem of Finite Abelian Groups; he moved that theorem, its corollaries, and the \(\gf(2^4)\) example from Finite Fields into the Groups chapter to go with it. In the Rings chapter, he rewrote the proof of the First Isomorphism Theorem for rings to be self-contained rather than invoking the (later) group-theoretic First Isomorphism Theorem, and trimmed several remaining references there to group theory.

A “Results Cited From Later Chapters” section was temporarily added to the Mathematical Prerequisites appendix to hold citations that did not yet fit naturally into the two new chapters or elsewhere. As those chapters absorbed everything it held, including the self-contained replacement above for its last entry, the Fundamental Theorem of Finite Abelian Groups, the section was removed entirely.

The original Sylow Theorems chapter was merged into Group Actions, replacing its Burnside’s Counting Theorem section. That chapter’s Groups Acting on Sets section was later moved into the new Groups chapter, and what remained (The Class Equation and The Sylow Theorems) was renamed The Sylow Theorems.

The Applications section of Galois Theory (Solvability by Radicals, Insolvability of the Quintic, and the Fundamental Theorem of Algebra) was split out into its own chapter, Applications of Galois Theory.

The book’s chapter order is now Rings, Polynomial Rings, Integral Domains, Vector Spaces, Extension Fields, Finite Fields, Groups, Galois Theory, The Simplicity of \(A_n\text{,}\) The Sylow Theorems, and Applications of Galois Theory.

The Preliminaries chapter was renamed Mathematical Prerequisites, and it and The Integers were moved to the back of the book as appendices.

The “Polynomial Codes” section (and its BCH Codes subsection) was removed from Finite Fields, along with every exercise tied to coding theory.

All Sage exercises, Sage tutorial sections, and Sage-related introductory material were removed throughout.

Reading Questions and Programming Exercises were removed from every chapter, to lighten the text.

The preface was replaced with a short piece on the significance of Galois’s work and how this text builds toward Galois theory as its capstone.

The title page was updated to reflect this edition, and the Sage Exercises and Spanish translation credits were removed.

The brand logo was replaced with a portrait of Évariste Galois, in the public domain via Wikimedia Commons.

The back matter’s umbrella title was renamed from “Reference” to “Back Matter,” now that it also holds appendix chapters, not just reference material.