Groups, Rings, Modules, Vector Spaces, and Fields
Tommy Chen
Version 1.14
Version 1.14 is a small maintenance update following Version 1.1, which
records the substantive revision of the finite-dimensional linear-algebra
material, particularly the expanded treatment of inner-product spaces. Small
maintenance updates continue the final decimal sequence: for example, 1.01
follows 1.0, while 1.11, 1.12, 1.13, and 1.14 follow 1.1. Larger
mathematical or editorial revisions advance the minor version, as in 1.1.
This update completes the contextual cleanup of dimensions and operator-space
notation, and adds one intermediate arithmetic step to the matrix-product
example.
This repository contains introductory graduate algebra notes that are self-contained in their algebraic development, with an elementary entry point and a graduate destination. Readers need strong high-school or early undergraduate mathematical maturity (sets, functions, elementary proofs, and the familiar number systems), but no prior abstract algebra. The notes develop definitions and proof techniques cumulatively, then prepare readers for commutative algebra, homological algebra, representation theory, algebraic geometry, and algebraic number theory.
Dummit--Foote, Abstract Algebra, 3rd ed., is the primary reference. Chapter 2 of Lecture Notes on Mathematical Analysis is optional preparation for readers seeking more practice with proof language, sets, functions, induction, and the familiar number systems. It is not a formal prerequisite: all algebraic material required in the manuscript is developed within these notes.
The manuscript has four completed parts.
- Groups develops groups, homomorphisms, quotients, actions, Sylow theory, semidirect products, composition series, solvability, and nilpotence.
- Rings and Polynomial Rings develops ideals, quotient rings, localization, Euclidean domains, PIDs, UFDs, polynomial factorization, and Gauss's Lemma.
- Modules and Linear Algebra develops modules, exact sequences, determinants, duality, bilinear and quadratic forms, PID modules, canonical forms, inner-product spaces, tensor products, and exterior algebra.
- Fields and Galois Theory develops field extensions, algebraic closures, embeddings, separability, normality, finite Galois theory, finite fields, cyclotomic extensions, radicals, direct and inverse limits, profinite groups, infinite Galois theory, and absolute Galois groups. It closes with the non-prerequisite survey Further Topics in Algebra.
The inner-product chapter is supplementary to the Dummit--Foote backbone and supplies finite-dimensional orthogonality, spectral theory, and singular-value preparation. Tensor products are developed over arbitrary rings through balanced maps and bimodules; arbitrary or infinite tensor products are intentionally excluded. A full classification theory of quadratic forms, detailed finite-group classification, and material after classical Galois theory are outside the stated scope. The classification of finitely generated abelian groups follows from the PID module structure theorem.
The finite-dimensional linear-algebra material has also been audited against Friedberg, Insel, and Spence, Linear Algebra, 5th ed. The principal revision strengthens the inner-product chapter with complete treatments of Cauchy--Schwarz, polarization, Bessel and Parseval, Gram--Schmidt, orthogonal projections, Riesz representation, adjoints, the spectral theorems, and singular-value decomposition. The existing vector-space, matrix, determinant, and canonical-form chapters received only targeted consistency checks. A standalone computational course in row reduction and systems of linear equations remains deliberately outside the notes' structural scope.
The manuscript uses complete proofs for results required later, consistent
tikz-cd diagrams, restrained borderless hyperlinks, and normalized notation.
Direct and inverse limits are introduced by universal properties without
explicit category or functor language.
The complex spectral theorem uses the Fundamental Theorem of Algebra as an external analytic input. Chapter 12 also gives a Galois-theoretic proof whose only substantive analytic ingredient is the Intermediate Value Theorem, together with elementary properties of real polynomials and real numbers. These facts are developed in Lecture Notes on Mathematical Analysis.
Chapter 12 includes the point-set-topology and topological-group bridge needed for profinite and infinite Galois theory. Tychonoff's theorem is used there as an external input; its full arbitrary-product form is noted to be equivalent over ZF to the Axiom of Choice.
Parts I--IV are mathematically complete and pedagogically frozen within the stated scope. Future changes are limited to typographical corrections or genuine mathematical errata.
Copyright © Tommy Chen. This work is licensed under the Creative Commons Attribution 4.0 International License.
The distinction between a theorem, example, and exercise in the primary reference is not a distinction in mathematical importance. A result needed by the notes' chosen scope must be stated and proved in the manuscript regardless of where Dummit--Foote places it. In particular, the notes never defer a subsequently used nontrivial result to the reader.
The operational standards and selection test are in EDITORIAL_POLICY.md. The evolving source and dependency record is coverage-audit.md.
References to Dummit--Foote must give the edition, section, and exercise number. Exercise numbers are intentionally not guessed: they are entered in the audit only after checking the edition used for the project. This avoids creating an apparently precise but unreliable cross-reference.
The LaTeX entry point is main.tex. It includes a detailed
introduction and twelve dependency-ordered chapters in chapters/. The project
uses theorem environments for definitions, results, examples, remarks, and
exercises, with complete proofs for all results on which later material depends.