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On methods of computing galois groups and their implementations in MAPLE.

by Tang Ko Cheung, Simon. / Thesis date on t.p. originally printed as 1997, of which 7 has been overwritten as 8 to become 1998. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 95-97). / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- Motivation --- p.5 / Chapter 1.1.1 --- Calculation of the Galois group --- p.5 / Chapter 1.1.2 --- Factorization of polynomials in a finite number of steps IS feasible --- p.6 / Chapter 1.2 --- Table & Diagram of Transitive Groups up to Degree 7 --- p.8 / Chapter 1.3 --- Background and Notation --- p.13 / Chapter 1.4 --- Content and Contribution of THIS thesis --- p.17 / Chapter 2 --- Stauduhar's Method --- p.20 / Chapter 2.1 --- Overview & Restrictions --- p.20 / Chapter 2.2 --- Representation of the Galois Group --- p.21 / Chapter 2.3 --- Groups and Functions --- p.22 / Chapter 2.4 --- Relative Resolvents --- p.24 / Chapter 2.4.1 --- Computing Resolvents Numerically --- p.24 / Chapter 2.4.2 --- Integer Roots of Resolvent Polynomials --- p.25 / Chapter 2.5 --- The Determination of Galois Groups --- p.26 / Chapter 2.5.1 --- Searching Procedures --- p.26 / Chapter 2.5.2 --- "Data: T(x1,x2 ,... ,xn), Coset Rcpresentatives & Searching Diagram" --- p.27 / Chapter 2.5.3 --- Examples --- p.32 / Chapter 2.6 --- Quadratic Factors of Resolvents --- p.35 / Chapter 2.7 --- Comment --- p.35 / Chapter 3 --- Factoring Polynomials Quickly --- p.37 / Chapter 3.1 --- History --- p.37 / Chapter 3.1.1 --- From Feasibility to Fast Algorithms --- p.37 / Chapter 3.1.2 --- Implementations on Computer Algebra Systems --- p.42 / Chapter 3.2 --- Squarefree factorization --- p.44 / Chapter 3.3 --- Factorization over finite fields --- p.47 / Chapter 3.4 --- Factorization over the integers --- p.50 / Chapter 3.5 --- Factorization over algebraic extension fields --- p.55 / Chapter 3.5.1 --- Reduction of the problem to the ground field --- p.55 / Chapter 3.5.2 --- Computation of primitive elements for multiple field extensions --- p.58 / Chapter 4 --- Soicher-McKay's Method --- p.60 / Chapter 4.1 --- "Overview, Restrictions and Background" --- p.60 / Chapter 4.2 --- Determining cycle types in GalQ(f) --- p.62 / Chapter 4.3 --- Absolute Resolvents --- p.64 / Chapter 4.3.1 --- Construction of resolvent --- p.64 / Chapter 4.3.2 --- Complete Factorization of Resolvent --- p.65 / Chapter 4.4 --- Linear Resolvent Polynomials --- p.67 / Chapter 4.4.1 --- r-sets and r-sequences --- p.67 / Chapter 4.4.2 --- Data: Orbit-length Partitions --- p.68 / Chapter 4.4.3 --- Constructing Linear Resolvents Symbolically --- p.70 / Chapter 4.4.4 --- Examples --- p.72 / Chapter 4.5 --- Further techniques --- p.72 / Chapter 4.5.1 --- Quadratic Resolvents --- p.73 / Chapter 4.5.2 --- Factorization over Q(diac(f)) --- p.73 / Chapter 4.6 --- Application to the Inverse Galois Problem --- p.74 / Chapter 4.7 --- Comment --- p.77 / Chapter A --- Demonstration of the MAPLE program --- p.78 / Chapter B --- Avenues for Further Exploration --- p.84 / Chapter B.1 --- Computational Galois Theory --- p.84 / Chapter B.2 --- Notes on SAC´ؤSymbolic and Algebraic Computation --- p.88 / Bibliography --- p.97

Identiferoai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_322226
Date January 1998
ContributorsTang, Ko Cheung Simon., Chinese University of Hong Kong Graduate School. Division of Mathematics.
Source SetsThe Chinese University of Hong Kong
LanguageEnglish
Detected LanguageEnglish
TypeText, bibliography
Formatprint, iv, 97 leaves ; 30 cm.
RightsUse of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

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