Abstract Algebra and Discrete Mathematics

By Karl Dahlke, Copyright © 2016

This e-book is an introduction to combinatorics, number theory, topology, groups, rings, fields, modules, algebraic and integral extensions, noncommutative algebra, algebraic number theory, algebraic geometry, algebraic topology, and even more. The focus is on breadth rather than depth. Excellent books already exist for any one of these topics in detail, and I don't want to reinvent that wheel. Instead, this book knits them all together, providing a foundation for each topic in turn. By analogy, you might point your backyard telescope to every corner of the galaxy, in an effort to comprehend its scope, beauty, and diversity. You might not understand the Crab Nebula in all its detail, but you should walk away with an appreciation for the vastness and the wonder of the galaxy, in this case, the galaxy of modern mathematics. If, from time to time, you find yourself saying, "How did anybody ever think of that?", then I have succeeded.

Each chapter builds on the information that has gone before, and forward references are rare, though they do happen from time to time. I hope this on-line book is more accessible than a sea of disconnected web pages, which is the hallmark of most math websites. Send along your questions or feedback.

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Like all self-respecting C programmers, I am numbering my chapters starting with 0.
  1. Prolog
  2. Data Structures
  3. Set Theory
  4. Combinatorics
  5. Number Theory
  6. Euclidean Domains
  7. Difference Equations
  8. Fields
  9. Finite Fields
  10. Cyclotomic Extensions
  11. Groups
  12. Groups Acting on Sets
  13. Similar Matrices and Jordan Canonical Form
  14. Finite Simple Groups
  15. Generating Functions
  16. Galois Groups and Extensions
  17. Straightedge Compass Construction
  18. Solvable Groups, Nilpotent Groups, and the Composition Series
  19. Solvable Extensions and Polynomials
  20. Separable Extensions
  21. Category Theory
  22. Free Groups
  23. Permutation Groups
  24. Rings and Ideals
  25. Principal Ideal Domains
  26. Modules
  27. Noetherian and Artinian Modules and Rings
  28. Modules over a PID
  29. Krull Schmidt
  30. Fractions
  31. Radical Ideals in a Commutative Ring
  32. Primary Ideals and Laskerian Rings
  33. Simple and Semisimple Rings and Modules
  34. The Jacobson Radical
  35. Radical Ideals in a Noncommutative Ring
  36. Local Rings
  37. Twisted Rings
  38. Division Rings
  39. Quadratic Forms
  40. Orbital Mechanics
  41. Matrix Polynomials
  42. Point Set Topology
  43. Metric Spaces
  44. Compact Sets
  45. Banach and Hilbert Spaces
  46. Topological Groups and Modules
  47. Topological Dimension
  48. Spec R
  49. Graded Rings
  50. Projective, Injective, Tensor Product
  51. R Algebras
  52. Integral Extensions
  53. Valuation Rings
  54. Dedekind Domains
  55. Elliptic Curves
  56. Lattice in n Space
  57. Integral Rings
  58. Algebraic Number Theory
  59. Quadratic Number Fields
  60. Continued Fractions
  61. Cyclotomic Number Fields
  62. Generalized Reciprocity
  63. Local Fields

What if you are interested in pythagorean triples, but you're not sure which chapter to look in? Number Theory is a good guess, but in fact those triples cannot be analyzed without the machinery of Euclidean Domains. Nobody wants to dip into each chapter, searching for a particular topic. The following master index presents every section of every chapter in order. Scroll through to get a feel for each chapter and what it contains, or use the control-F search function in your browser to look for particular keywords on this page, then jump straight to that section. Realize however that you may need to start reading at the top of the chapter for context and clarity.

Data Structures

Set Theory


Number Theory

Euclidean Domains

Difference Equations


Finite Fields

Cyclotomic Extensions


Groups Acting on Sets

Similar Matrices and Jordan Canonical Form

Finite Simple Groups

Generating Functions

Galois Groups and Extensions

Straightedge Compass Construction

Solvable Groups

Solvable Extensions and Polynomials

Separable Extensions

Category Theory

Free Groups

Permutation Groups

Rings and Ideals

Principal Ideal Domains


Noetherian and Artinian Modules and Rings

Modules over a PID

Krull Schmidt


Radical Ideals in a Commutative Ring

Primary Ideals and Laskerian Rings

Simple and Semisimple Rings and Modules

The Jacobson Radical

Radical Ideals in a Noncommutative Ring

Local Rings

Twisted Rings

Division Rings

Quadratic Forms

Orbital Mechanics

Matrix Polynomials

Point Set Topology

Metric Spaces

Compact Sets

Banach and Hilbert Spaces

Topological Groups and Modules

Topological Dimension

Spec R

Graded Rings

Projective, Injective, Tensor Product

R Algebras

Integral Extensions

Valuation Rings

Dedekind Domains

Elliptic Curves

Lattice in n Space

Integral Rings

Algebraic Number Theory

Quadratic Number Fields

Continued Fractions

Cyclotomic Number Fields

Generalized Reciprocity

Local Fields