Thesis (S.M.)--University of Chicago, Department of Mathematics, 1908. / Includes bibliographical references. Also available on the Internet.

Thesis (M.A.) in Mathematics--University of Maine, 2009. / Includes vita. Includes bibliographical references (leaf 28).

Thesis (Honors)--Liberty University Honors Program, 2007. / Includes bibliographical references. Also available through Liberty University's Digital Commons.

No description available.

by Song Li-Min. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Bibliography: leaves 175-178.

This thesis is devoted to application of topological ideas to Galois theory. In the fi rst part we obtain a characterization of branching data that guarantee that a regular mapping from a Riemann surface to the Riemann sphere having this branching data is invertible in radicals. The mappings having such branching data are then studied with emphasis on those exceptional properties of these mappings that single them out among all mappings from a Riemann surface to the Riemann sphere. These results provide a framework for understanding an earlier work of Ritt on rational functions invertible in radicals. In the second part of the thesis we apply topological methods to prove lower bounds in Klein's resolvent problem, i.e. the problem of determining whether a given algebraic function of n variables is a branch of a composition of rational functions and an algebraic function of k variables. The main topological result here is that the smallest dimension of the base-space of a covering from which a given covering over a torus can be induced is equal to the minimal number of generators of the monodromy group of the covering over the torus. This result is then applied for instance to prove the bounds k is at least n/2 in Klein's resolvent problem for the universal algebraic function of degree n and the answer k = n for generic algebraic function of n variables of degree at least 2n.

After the new impulse given to the theory of algebraic equations by the discoveries of Lagrange and Vandermonde in 1770, Ruffini tried to solve the problem where Lagrange had left it, i.e., proved the impossibility of solving by radicals the general equation of the fifth degree. His proof still remains unclear but nevertheless is very similar to the proof obtained by Abel later. Looking for new types of equations solvable by radicals, the latter reached the conception of "abelian" extensions and showed the solvability by radicals in this case. He defined the notion of irreducible polynomials over a given field. After him, Galois defined what was to be called the Galois group of a polynomial and showed that a polynomial is solvable by radicals if its Galois group is solvable. [...]

This thesis is devoted to application of topological ideas to Galois theory. In the fi rst part we obtain a characterization of branching data that guarantee that a regular mapping from a Riemann surface to the Riemann sphere having this branching data is invertible in radicals. The mappings having such branching data are then studied with emphasis on those exceptional properties of these mappings that single them out among all mappings from a Riemann surface to the Riemann sphere. These results provide a framework for understanding an earlier work of Ritt on rational functions invertible in radicals. In the second part of the thesis we apply topological methods to prove lower bounds in Klein's resolvent problem, i.e. the problem of determining whether a given algebraic function of n variables is a branch of a composition of rational functions and an algebraic function of k variables. The main topological result here is that the smallest dimension of the base-space of a covering from which a given covering over a torus can be induced is equal to the minimal number of generators of the monodromy group of the covering over the torus. This result is then applied for instance to prove the bounds k is at least n/2 in Klein's resolvent problem for the universal algebraic function of degree n and the answer k = n for generic algebraic function of n variables of degree at least 2n.

Thesis (M.S.)--Oregon State College, 1952. / Typescript. Includes bibliographical references (leaf 21). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Missouri, 1935. / Vita. Photolithoprinted. Includes bibliographical references (leaf following p. 40).