Part I. Basics

1.Algebraic Themes

What is Symmetry? Symmetries of the Rectangle and the Square. Multiplication Tables. Symmetries and Matrices. Permutations. Divisibility in the Integers. Modular Arithmetic. Polynomials. Counting. Groups. Rings and Fields. An Applicaton to Cryptography.

2. Basic Theory of Groups.

First Results. Subgroups and Cyclic Groups. The Dihedral Groups. Homomorphisms and Isomorphisms. Cosets and Lagrange's Theorem. Equivalence Relations and Set Partitions. Quotient Groups and Homomorphism Theorems.

3. Products of Groups

Direct Products. Semidirect Products. Finite Abelian Groups. Vector Spaces.

4. Symmetries of Polyhedra.

Rotations of Regular Polyhedra. Rotation Groups of the Dodecahedron and Icosahedron. What about Reflections? Linear Isometries. The Full Symmetry Group and Chirality.

5. Actions of Groups.

Group Actions on Sets. Group Actions -- Counting Orbits. Symmetries of Groups. Group Actions and Group Structure. Application: Transitive Subgroups of S_5. Additional Exercises for Chapter 5.

7. Rings.

A Recollection of Rings. Homomorphisms and Ideals. Quotient Rings. INtegral Domains. Euclidean Domains, Principal Ideal Domains and Unique Factorization. Unique Factorization Domains. Noetherian Rings. Irreducibility Criteria.

8. Field Extensions -- First Look.

A Brief History. Solving the Cubic Equatiion. Adjoining Algebraic Elements to a Field. Splitting Field of a Cubic Polynomial. Splitting Field of Polynomials with Complex Coefficients.

Part II. Topics.

9. Field Extensions -- Second Look.

Finite and Algebraic Extensions. Splitting Fields. The Derivative and Multiple Roots. Splitting Fields and Automorphisms. The Galois Correspondence. Symmetric Functions. The General Equation of Degree n. Quartic Polynomials. Galois Groups of Higher Degree Polynomials.

10. Solvability.

Composition Series and Solvable Groups. Commutators and Solvability. Simplicity of the Alternating Groups. Cyclotomic Polynomials. Nth roots. Solvability by Radicals. Radical Extensions.

11. Isometry Groups.

More on Isometries of Euclidean Space. Euler's Theorem. Finite Rotation Groups. Crystals.

Appendix A. Almost Enough about Logic.

Appendix B. Almost Enough about Sets.

Appendix C. Induction.

Appendix D. Complex Numbers.

Appendix E. Review of Linear Algebra.

Appendix F. Models of Regular Polyhedra.

Appendix G. Suggestions for Further Study.

Index.