In 1993, Muzychuk showed that the rational S-rings over a cyclic group Z_n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z_n. This idea is easily extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational S-rings over G in a natural way. Our main result is that any finite group may be represented as the automorphism group of such a rational S-ring over an abelian p-group. In order to show this, we first give a complete description of the automorphism classes and characteristic subgroups of finite abelian groups. We show that for a large class of abelian groups, including all those of odd order, the lattice of characteristic subgroups is distributive. We also prove a converse to the well-known result of Muzychuk that two S-rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic S-rings. Finally, we show that the automorphism group of any S-ring over a cyclic group is abelian.

In Part I of the thesis an account is given of the basic algebra of extension fields which is required for the understanding of Galois theory. The fundamental theorem states the relationships of the subgroups of a permutation group of the root field of an equation to the subfields which are left invariant by these subgroups. Extensions of the basic theorem conclude Part I. In part II the solvability of equations by radicals is discussed, for fields of characteristic zero. A discussion of finite fields and primitive roots leads to a criterion for the solvability by radicals of equations over fields of prime characteristic. Finally, a method for determining the Galois group of any equation is discussed. Most of the material in the introductory chapters is taken from Artin's: Galois Theory (cf. p. 120). / Thesis / Master of Arts (MA)

General uncertainty analysis was used to evaluate the performance of a Rocket-Based Combined Cycle (RBCC) engine system. To estimate the uncertainties of test results, uncertainties of basic measurements such as temperature, pressure, mass flow rate, and thrust were determined. The desired test results of interest included specific impulse and characteristic velocity. Various possible test facilities were reviewed to obtain background information and example test run conditions. Based on the test run conditions, five methods of determining specific impulse were evaluated. Also, theoretical and actual characteristic velocities were analyzed to evaluate C* efficiency. Initially, general uncertainty analyses were completed relative to 1% accuracy for each measured variable. Then, cases were run using more realistic uncertainty estimates. The relative contributions of the different variables? uncertainties to the overall uncertainty of the selected performance parameters were also calculated. This process helps to identify the critical measurements from an uncertainty standpoint and can be a significant guide in the cost effective use of resources to reduce the test uncertainty.

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