In this dissertation the question of whether or not a relative extension of number fields has a relative integral basis is considered. In Chapters 2 and 3 we use a criteria of Mann to determine when a cyclic quartic field or a pure quartic field has an integral basis over its quadratic subfield. In the final chapter we study the question: if the relative discriminant of an extension K / k is principal, where [K : k] = l such that l is an odd prime and k is either a quadratic or a normal quartic number field, does K / k have an integral basis? / Ph. D.

A well-known theorem in "Number Theory", the Chinese Remainder Theorem, was first utilized by Paul K. Lin in constructing confounding systems for mixed factorial experiments. This study extends the use of the theorem to cover cases when more than one component from some of the symmetrical factorials are confounded, and to include cases where the number of levels of factors are not all relative prime. The second part of this study concerns the randomized fractional replicates, a procedure which selects confounded subsets with pre-assigned probabilities. This procedure provides full information on a specific set of parameters of interest while making no assumption of zero nuisance parameters. Estimation procedures in general symmetrical as well as asymmetrical factorial systems are studied under a ”fully orthogonalized" model. The type-g estimator, investigated under the generalized inverse operator, and the class of linear estimators of parameters of interest and their variance-covariance matrices are given. The unbiasedness of these estimators can be obtained only under the condition that each subset of treatment combinations is selected with equal probability. This work is concluded with simulation studies to compare the classical and the randomization procedures. The results indicate that when information about the nuisance parameters is not available, randomization procedure guards against a bad choice of design. / Ph. D.