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Computing splitting fields and galois groups over local function fields / by Anthoula Zervou ; Supervisor: Prof. Dr. Jürgen Klüners. Paderborn, 2024
Inhalt
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
1.9.1 Computation of Galois Groups
1.10 Ramification Groups
2 Ramification Polygon & Applications
2.1 Ramification Polygons
2.1.1 Computation of Ramification Polygons
2.2 Residual Polynomials
2.2.1 Computation of Residual Polynomials
2.3 Splitting Fields and Galois Groups in the One Segment Case
2.3.1 Splitting Fields in the One Segment Case
2.3.2 Galois Groups in the One Segment Case
2.4 Reduction to p-Extensions
3 Splitting Fields over Local Function Fields
3.1 Basic Approach
3.2 A Variation of the Basic Approach
3.2.1 Precision
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
4.5.1 Computation without Denominators
4.5.2 Computation with Denominators
4.6 Examples
5 Splitting Fields: Combined Approach
5.1 Splitting Fields in the General Case
5.1.1 The Splitting Field Algorithm
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
6.1 Automorphisms as Permutations
6.2 Galois Groups in the Last Step
6.3 Galois Groups
6.4 Examples
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
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