Computing splitting fields and galois groups over local function fields / by Anthoula Zervou ; Supervisor: Prof. Dr. Jürgen Klüners. Paderborn, 2024
Content
- Acknowledgments
- Abstract
- Contents
- Introduction
- 1 Preliminaries
- 1.1 Local Fields
- 1.2 Local Function Fields
- 1.3 Basic Galois Theory
- 1.4 Extensions of Local Fields
- 1.5 Unramified Extensions
- 1.6 Totally Ramified Extensions
- 1.7 Tamely Ramified Extensions
- 1.8 Newton Polygons
- 1.9 Splitting Fields and Galois Groups of Tamely Ramified Extensions
- 1.10 Ramification Groups
- 2 Ramification Polygon & Applications
- 2.1 Ramification Polygons
- 2.2 Residual Polynomials
- 2.3 Splitting Fields and Galois Groups in the One Segment Case
- 2.4 Reduction to p-Extensions
- 3 Splitting Fields over Local Function Fields
- 4 Factorization
- 4.1 Basic Symmetric Functions and Higher Traces of Roots
- 4.2 Fast Conversion between Polynomials and Higher Traces of Roots
- 4.2.1 From Polynomials to Newton Representations
- 4.2.2 From Newton Representations to Polynomial Representations
- 4.3 The Arbitrary Positive Characteristic Case
- 4.4 Minimal Polynomials of Linear Recurrent Sequences
- 4.5 Defining Polynomials of Extensions
- 4.6 Examples
- 5 Splitting Fields: Combined Approach
- 5.1 Splitting Fields in the General Case
- 5.2 Efficiency of the Combined Approach
- 5.3 Examples: Comparison of Version 1 to Version 2
- 5.4 Improvement of Splitting Field Computations
- 5.5 Examples: Comparison of Version 2 to Version 3
- 5.6 Reducible Polynomials
- 6 Computation of Galois Groups over Local Function Fields
- A Examples
- A.1 Splitting Fields: Comparison of Version 1 to Version 2
- A.2 Splitting Fields: Comparison of Version 2 to Version 3
- A.3 Splitting Fields and Galois Groups
- A.4 Factorization Code
- References
- List of Figures
- List of Algorithms
- Notation