Abstract Algebra and Discrete Mathematics
This e-book is an introduction to
groups, rings, fields, modules,
algebraic and integral extensions,
algebraic number theory,
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.
- 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, Nilpotent Groups, and the Composition Series
- 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
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.
Groups Acting on Sets
Similar Matrices and Jordan Canonical Form
Finite Simple Groups
Galois Groups and Extensions
Straightedge Compass Construction
Solvable Extensions and Polynomials
Rings and Ideals
Principal Ideal Domains
Noetherian and Artinian Modules and Rings
Modules over a PID
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
Point Set Topology
Banach and Hilbert Spaces
Topological Groups and Modules
Projective, Injective, Tensor Product
Lattice in n Space
Algebraic Number Theory
Quadratic Number Fields
Cyclotomic Number Fields