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

• Other Authors: Tang, Ko Cheung Simon.
• Format: Others
• Language: English
• Published: [199
• Subjects: Polynomials > Data processing; Galois theory; Group theory
• Online Access: http://library.cuhk.edu.hk/record=b5896319; http://repository.lib.cuhk.edu.hk/en/item/cuhk-322226

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