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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.

The Egyptian Fractions of One problem (EFO), asks the following question: Given a positive integer n, how many ways can 1 be expressed as the sum of n non-increasing unit fractions? In this paper, we verify a result concerning the EFO problem for n=8, and show the computational complexity of the problem can be severely lessened by new theorems concerning the structure of solutions to the EFO problem. / Master of Science / Expressing numbers as fractions has been the subject of one’s education since antiquity. This paper shows how we can write the number 1 as the sum of uniquely behaved fractions called “unit fractions”, that is, fractions with 1 in the numerator and some natural counting number in the denominator. Counting the number of ways this can be done reveals certain properties about the prime numbers, and how they interact with each other, as well as pushes the boundaries of computing power.

This dissertation is a study of two basic questions involving irreducible elements in algebraic number fields. The first question is: Given an algebraic integer β in a field with class number greater than two, how many different lengths of factorizations into irreducibles exist? The distribution into ideal classes of the prime ideals whose product is the principal ideal (β) determines the possible length of the factorizations into irreducibles. Chapter 2 gives precise answers when the field has class number 3 or 4, as well as when the class group is an elementary 2-group of order 8. The second question is: In a normal extension, when are there rational primes which split completely and remain irreducible? Chapter 3 focusses on the bicyclic bi-quadratic fields. The imaginary bicyclic biquadratic fields which contain such primes are completely determined. / 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.