A First Course in Abstract Algebra

A Few Preliminaries.


Mathematics and Proofs.



Sets and Relations.



Mathematical Induction.



Complex and Matrix Algebra.



1. Groups and Subgroups.


Binary Operations.



Finite-State Machines (Automata).



Isomorphic Binary Structures.



Groups.



Subgroups.



Cyclic Groups and Generators.



Cayley Digraphs.



2. More Groups and Cosets.


Groups of Permutations.



Automata.



Orbits, Cycles, and the Alternating Groups.



Plane Isometries.



Cosets and the Theorem of Lagrange.



Direct Products and Finitely Generated Abelian Groups.



Periodic Functions, Plane Isometries.



Binary Linear Codes.



3. Homomorphisms and Factor Groups.


Homomorphisms.



Factor Groups.



Factor-group Computations and Simple Groups.



Series of Groups.



Group Action on a Set.



Applications of G-sets to Counting.



4. Advanced Group Theory.


Isomorphism Theorems; Proof of the Jordan-Holder Theorem.



Sylow Theorem.



Applications of the Sylow Theory.



Free Abelian Groups.



Free Groups.



Group Presentations.



5. Introduction to Rings and Fields.


Rings and Fields.



Integral Domains.



Fermat's and Euler's Theorems.



The Field of Quotients of an Integral Domain.



Rings of Polynomials.



Factorization of Polynomials over a Field.



Noncommutative Examples.



Ordered Rings and Fields.



6. Factor Rings and Ideals.


Homomorphisms and Factor Rings.



Prime and Maximal Ideals.



Gröbner Bases for Ideals.



7. Factorization*.


Unique Factorization Domains.



Euclidean Domains.



Gaussian Integers and Norms.



8. Extension Fields.


Introduction to Extension Fields.



Vector Spaces.



Algebraic Extensions.



Geometric Constructions.



Finite Fields.



Additional Algebraic Structures.



9. Automorphisms and Galois Theory.


Automorphisms of Fields.



The Isomorphism Extension Theorem.



Splitting Fields.



Separable Extensions.



Totally Inseparable Extensions.



Galois Theory.



Illustrations of Galois Theory.



Cyclotomic Extensions.



Insolvability of the Quintic.



Bibliography.


Notations.


Answers To Odd-Numbered Exercises Not Requiring Proofs.


Index.


*Not required for the remainder of the text.