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Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Undergraduate honors paper--Mount Holyoke College, 2006. Dept. of Mathematics. / Includes bibliographical references (leaves 66-67).

For definite quadratic lattices over the rings of integers of algebraic number fields, it is shown that lattices are determined up to isometry by their local structure and sublattices of codimension 1. In particular, a theorem of Yoshiyuki Kitaoka for $\mathbb{Z}$-lattices is generalized to definite lattices over algebraic number fields.

Fung Yiu-cho. / Bibliography: leaves 93-114 / Thesis (M.Phil.)--Chinese University of Hong Kong, 1982

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Thesis (Ph. D.)--McMaster University (Canada), 1996. / Source: Dissertation Abstracts International, Volume: 58-06, Section: B, page: 3074. Adviser: V.P. Snaith.

Thesis (Ph. D.)--University of Texas at Austin, 2005. / Vita. Includes bibliographical references.

In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this dissertation, we re-examine Eisenstein congruences, incorporating a notion of “depth of congruence,” in order to understand the local structure of Eisenstein ideals associated to weight 2 cusp forms of squarefree level N. Specifically, we use a commutative algebra result of Berger, Klosin, and Kramer to bound the depth of mod p Eisenstein congruences (from below) by the p-adic valuation of φ(N). We then show how this depth of congruence controls the local principality of the associated Eisenstein ideal.

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.