Thesis (Ph. D.)--University of Wisconsin--Madison, 1940. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (M.A.)--The University of North Carolina at Greensboro, 2008. / Directed by Paul Duvall; submitted to the Dept. of Mathematical Sciences. Title from PDF t.p. (viewed Aug. 25, 2009). Includes bibliographical references (p. 35-36).

Thesis (Ph. D.) -- California Institute of Technology, 2010. / Title from home page (viewed 06/21/2010). Advisor and committee chair names found in the thesis' metadata record in the digital repository. Includes bibliographical references.

The objective of this dissertation was to determine the class of domains that are both almost Dedekind and atomic. To investigate this question we constructed a global object called the norm, and used it to determine properties that a domain must have to be both atomic and almost Dedekind. Additionally we use topological notions on the spectrum of a domain to determine atomicity. We state some theorems with regard to ACCP and class groups. The lemmas and theorems in this dissertation answer in part the objective. We conclude with a chapter of future study that aims to approach a complete answer to the objective.

<p> This thesis makes a small study of various kinds of submonoids of abelian monoids, of the lattice of submonoids of an abelian monoid, and of the concept of unique factorization in abelian monoids. Also, the main concepts in category theory are examined in the special case of the category of abelian monoids and some of its subcategories. </p> / Thesis / Master of Science (MSc)

by Li King Hung. / Thesis submitted in: July, 1998. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 54-56). / Abstract also in Chinese. / Chapter 1 --- Description of the Number Field Sieve --- p.6 / Chapter 1.1 --- Introduction --- p.6 / Chapter 1.2 --- Outline of the algorithm --- p.8 / Chapter 2 --- Algebraic knowledge --- p.17 / Chapter 2.1 --- Factorization of an ideal over the class of ideals --- p.17 / Chapter 2.2 --- Existence of the square roots of an element in the ring of algebraic integers --- p.26 / Chapter 3 --- Run time analysis and Practical result --- p.31 / Chapter 3.1 --- Relation between sieving over X and finding the linear dependencies --- p.32 / Chapter 3.2 --- Relation between the size of a factor base and finding the linear dependencies --- p.34 / Chapter 3.3 --- Practical consideration --- p.36 / Chapter 4 --- Improvement of the algorithm --- p.38 / Chapter 4.1 --- Quadratic characters --- p.38 / Chapter 4.2 --- Finding the square root --- p.40 / Chapter 4.3 --- Solving the linear system of equation --- p.42 / Chapter 4.4 --- Reusing the computation --- p.47 / Chapter 4.5 --- Using more general purpose data --- p.50 / Chapter 4.6 --- Examples --- p.51 / Chapter 4.6.1 --- "A 18-digit example,761260375069630873" --- p.52 / Chapter 4.6.2 --- "A 23-digit example, 16504377514594481520559" --- p.52 / Bibliography

In [2], Dan Anderson and Andrea Frazier introduced a generalized theory of factorization. Given a relation τ on the nonzero, nonunit elements of an integral domain D, they defined a τ-factorization of a to be any proper factorization a = λa1 · · · an where λ is in U (D) and ai is τ-related to aj, denoted ai τ aj, for i not equal to j . From here they developed an abstract theory of factorization that generalized factorization in the usual sense. They were able to develop a number of results analogous to results already known for usual factorization. Our work focuses on the notion of τ-factorization when the relation τ has characteristics similar to those of coprimeness. We seek to characterize such τ-factorizations. For example, let D be an integral domain with nonzero, nonunit elements a, b ∈ D. We say that a and b are comaximal (resp. v-coprime, coprime ) if (a, b) = D (resp., (a, b)v = D, [a, b] = 1). More generally, if ∗ is a star-operation on D, a and b are ∗-coprime if (a, b)∗ = D. We then write a τmax b (resp. a τv b, a τ[ ] b, or a τ∗ b) if a and b are comaximal (resp. v -coprime, coprime, or ∗-coprime).

We consider and solve some open conjectures on the asymptotic behavior of the number of different numbers of the nonassociate factorizations of prescribed minimal length for specific finite factorization domains. The asymptotic behavior will be classified for Cohen-Kaplansky domains in Chapter 1 and for domains of the form R=K+XF[X] for finite fields K and F in Chapter 2. A corollary of the main result in Chapter 3 will determine the asymptotic behavior for Krull domains with finite divisor class group.

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