Abstract Algebra and Famous Impossibilities

0.1 Three Famous Problems.- 0.2 Straightedge and Compass Constructions.- 0.3 Impossibility of the Constructions.- 1 Algebraic Preliminaries.- 1.1 Fields, Rings and Vector Spaces.- 1.2 Polynomials.- 1.3 The Division Algorithm.- 1.4 The Rational Roots Test.- Appendix to Chapter 1.- 2 Algebraic Numbers and Their Polynomials.- 2.1 Algebraic Numbers.- 2.2 Monic Polynomials.- 2.3 Monic Polynomials of Least Degree.- 3 Extending Fields.- 3.1 An Illustration: $$\mathbb{Q}(\sqrt 2 )$$.- 3.2 Construction of $$\mathbb{F}(\alpha )$$.- 3.3 Iterating the Construction.- 3.4 Towers of Fields.- 4 Irreducible Polynomials.- 4.1 Irreducible Polynomials.- 4.2 Reducible Polynomials and Zeros.- 4.3 Irreducibility and irr$$(\alpha ,\mathbb{F})$$.- 4.4 Finite-dimensional Extensions.- 5 Straightedge and Compass Constructions.- 5.1 Standard Straightedge and Compass Constructions.- 5.2 Products, Quotients, Square Roots.- 5.3 Rules for Straightedge and Compass Constructions.- 5.4 Constructible Numbers and Fields.- 6 Proofs of the Impossibilities.- 6.1 Non-Constructible Numbers.- 6.2 The Three Constructions are Impossible.- 6.3 Proving the "All Constructibles Come From Square Roots" Theorem.- 7 Transcendence of e and ?.- 7.1 Preliminaries.- 7.2 e is Transcendental.- 7.3 Preliminaries on Symmetric Polynomials.- 7.4 ? is Transcendental - Part 1.- 7.5 Preliminaries on Complex-valued Integrals.- 7.6 ? is Transcendental - Part 2.- 8 An Algebraic Postscript.- 8.1 The Ring $$\mathbb{F}\left[ X \right]_{p(X)}$$.- 8.2 Division and Reciprocals in $$\mathbb{F}\left[ X \right]_{p(X)}$$.- 8.3 Reciprocals in $$\mathbb{F}\left( \alpha \right)$$.- 9 Other Impossibilities and Abstract Algebra.- 9.1 Construction of Regular Polygons.- 9.2 Solution of Quintic Equations.- 9.3 Integration in Closed Form.