Author | Jones, Arthur. author |
---|---|
Title | Abstract Algebra and Famous Impossibilities [electronic resource] / by Arthur Jones, Kenneth R. Pearson, Sidney A. Morris |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1991 |
Connect to | http://dx.doi.org/10.1007/978-1-4419-8552-1 |
Descript | X, 189 p. online resource |
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 -- 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 โAll Constructibles Come From Square Rootsโ Theorem -- 7 Transcendence of e and ? -- 7.1 Preliminaries -- 7.2 e is Transcendental -- 7.3 Preliminaries on Symmetric Polynomials -- 7.4 ? is Transcendental โ Part 1 -- 7.5 Preliminaries on Complex-valued Integrals -- 7.6 ? is Transcendental โ 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