Geometry of the Quintic

Contents
Preface
Chapter 1. The complex sphere
1 Topological preliminaries
2 Stereographic projection
3 The Riemann sphere and meromorphic functions
4 The complex projective line and algebraic mappings
5 Summary
Chapter 2. Finite automorphism groups of the sphere
1 Automorphisms
2 Rotations of the Riemann sphere
3 Finite automorphism groups and rotation groups
4 Group actions
5 The Platonic solids and their rotations
6 Finite rotation groups of the sphere
7 Projective representations of the finite rotation groups
8 Summary
Chapter 3. Invariant functions
1 Invariant forms
2 Orbit-forms and invariant forms
3 Covariant forms
4 Calculation of the degenerate orbit-forms
5 Invariant algebraic mappings
6 Invariant rational functions
7 Summary
Chapter 4. Inverses of the invariant functions
1 Fields and polynomials
2 Algebraic extensions
3 Galois extensions
4 The rotation group extension
5 The Radical Criterion
6 Algebraic inversion of the nonicosahedral invariants
7 Resolvents
8 The Brioschi resolvent
9 Inversion of the icosahedral invariant
10 Summary
Chapter 5. Reduction of the quintic to Brioschi form
1 The general polynomial extension
2 Newton's identities
3 Resultants
4 Tschirnhaus transformations and principal form
5 Galois theory of the Tschirnhaus transformation
6 Projective space and algebraic sets
7 Geometry of the Tschirnhaus transformation
8 Brioschi form
9 Summary
Chapter 6. Kronecker's Theorem
1 Transcendence degree
2 Kronecker's Theorem
3 Lüroth's Theorem
4 The Embedding Lemma
5 Summary
Chapter 7. Computable extensions
1 Newton's method for nth roots
2 Varieties and function fields
3 Purely iterative algorithms
4 Iteratively constructible extensions
5 Differential forms
6 Normal rational functions
7 Ingredients of the algorithm
8 General convergence of the model
9 Computing the algorithm
10 Solving the Brioschi quintic by iteration
11 Onward
Index
List of symbols
Bibliography