Abstract Algebra Book

1. Introduction. 0. Background. Part I: GEOMETRY. 0. What is Congruence? 1. Some 2-Dimensional Geometry. 2. Symmetry. Part II: POLYNOMIALS. 3. The Root of It All. 4. The Renaissance of Algebra. 5. Complex Numbers. 6. Symmetric Polynomials and the Fundamental Theorem of Algebra. 7. Permutations and LaGranges's Theorem. 8. Orbits and Cauchy's Formula. 9a. Hamilton's Quaternions (optional). Part III: NUMBERS. 10. Back to Euclid. 11. Euclid's Lemma for Polynomials. 12. Fermat and the Rebirth of Number Theory. 13. LaGrange's Theorem Revisited. 14. Rings and Squares. 15. Fermat's Last Theorem (for Polynomials). 15a. Still More Fermat's Last Theorem (optional). Part IV: THE GRAND SYNTHESIS. 16. Constructible Polygons and the Method of Mr. Gauss. 17. Cyclotomic Fields and Linear Algebra. 18. A LaGrange Theorem for Fields and Non-Constructibility. 19. Galois Fields and the Fundamental Theorem of Algebra Revisited. 20. Galois' Theory of Equations. 21. The Galois Correspondence. 22. Constructible Numbers and Solvable Equations. 23. Glossary of Important Terms.