Numbers and Symmetry: An Introduction to Algebra

Chapter 1. New Numbers A Planeful of Integers, Z[i] Circular Numbers, Zn More Integers on the Number Line, Z [v2] Notes Chapter 2. The Division Algorithm Rational Integers Norms Gaussian Numbers Q (v2) Polynomials Notes Chapter 3. The Euclidean Algorithm Bezout's Equation Relatively Prime Numbers Gaussian Integers Notes Chapter 4. Units Elementary Properties Bezout's Equations Wilson's Theorem Orders of Elements: Fermat and Euler Quadratic Residues Z [v2] Notes Chapter 5. Primes Prime Numbers Gaussian Primes Z [v2] Unique Factorization into Primes Zn Notes Chapter 6. Symmetries Symmetries of Figures in the Plane Groups The Cycle Structure of a Permutation Cyclic Groups The Alternating Groups Notes Chapter 7. Matrices Symmetries and Coordinates Two-by-Two Matrices The Ring of Matrices M2(R) Units Complex Numbers and Quaternions Notes Chapter 8. Groups Abstract Groups Subgroups and Cosets Isomorphism The Group of Units of a Finite Field Products of Groups The Euclidean Groups E (1), E (2), and E (3) Notes Chapter 9. Wallpaper Patterns One-Dimensional Patterns Plane Lattices Frieze Patterns Space Groups The 17 Plane Groups Notes Chapter 10. Fields Polynomials Over a Field Kronecker's Construction of Simple Field Extensions Finite Fields Notes Chapter 11. Linear Algebra Vector Spaces Matrices Row Space and Echelon Form Inverses and Elementary Matrices Determinants Notes Chapter 12. Error-Correcting Codes Coding for Redundancy Linear Codes Parity-Check Matrices Cyclic Codes BCH Codes CDs Notes Chapter 13. Appendix: Induction Formulating the n-th Statement The Domino Theory: Iteration Formulating the Induction Statement Squares Templates Recursion Notes Chapter 14. Appendix: The Usual Rules Rings Notes Index