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Thesis (Ph. D.)--University of Chicago, / Includes bibliographical references. Also available on the Internet.

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This paper will be devoted to an exposition of some of the relationships existing between a field and certain of its extension fields. In particular, it will be shown that many fields may be characterized rather simply in terms of their subfields which, in turn, may be directly correlated with the subgroups of a finite group of automorphisms of the given field.

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

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