Preliminaries Integer Factorization Functions Modular Arithmetic Rational and Real Numbers Understanding the Group Concept Introduction to Groups Modular Congruence The Definition of a Group The Structure within a Group Generators of Groups Defining Finite Groups in Sage Subgroups Patterns within the Cosets of Groups Left and Right Cosets Writing Secret Messages Normal Subgroups Quotient Groups Mappings between Groups Isomorphisms Homomorphisms The Three Isomorphism Theorems Permutation Groups Symmetric Groups Cycles Cayley's Theorem Numbering the Permutations Building Larger Groups from Smaller Groups The Direct Product The Fundamental Theorem of Finite Abelian Groups Automorphisms Semi-Direct Products The Search for Normal Subgroups The Center of a Group The Normalizer and Normal Closure Subgroups Conjugacy Classes and Simple Groups The Class Equation and Sylow's Theorems Solvable and Insoluble Groups Subnormal Series and the Jordan-Hölder Theorem Derived Group Series Polycyclic Groups Solving the PyraminxTM Introduction to Rings The Definition of a Ring Entering Finite Rings into Sage Some Properties of Rings The Structure within Rings Subrings Quotient Rings and Ideals Ring Isomorphisms Homomorphisms and Kernels Integral Domains and Fields Polynomial Rings The Field of Quotients Complex Numbers Ordered Commutative Rings Unique Factorization Factorization of Polynomials Unique Factorization Domains Principal Ideal Domains Euclidean Domains Finite Division Rings Entering Finite Fields in Sage Properties of Finite Fields Cyclotomic Polynomials Finite Skew Fields The Theory of Fields Vector Spaces Extension Fields Splitting Fields Galois Theory The Galois Group of an Extension Field The Galois Group of a Polynomial in Q The Fundamental Theorem of Galois Theory Applications of Galois Theory Appendix: Sage vs. Mathematica® Answers to Odd-Numbered Problems Bibliography.