No description available.

Thesis (Ph. D.)--University of Chicago, 1936. / Vita. Lithoprinted. "Private edition, distributed by the University of Chicago libraries, Chicago, Illinois."

The main goal of this Masters' thesis is to explore isomorphism types of quaternion algebras using the theory of quadratic forms, number theory and algebra. I would also present ways to characterize quaternion algebras, and talk about how quaternion algebras are important in Brauer groups by describing a theorem proved by Merkurjev in 1981.

The main goal of this Masters' thesis is to explore isomorphism types of quaternion algebras using the theory of quadratic forms, number theory and algebra. I would also present ways to characterize quaternion algebras, and talk about how quaternion algebras are important in Brauer groups by describing a theorem proved by Merkurjev in 1981.

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.

Orientador: Antonio Jose Engler / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-06T01:37:33Z (GMT). No. of bitstreams: 1 OliveiraJunior_MauroRibeirode_M.pdf: 283417 bytes, checksum: ed6ca467b6f01b1fd631b10cca398846 (MD5) Previous issue date: 2006 / Resumo: Neste trabalho seguimos, nos quatro primeiros capítulos, para a construção efetiva de extensões quaterniônicas, a partir do acúmulo de informações obtidas nos capítulos iniciais 1 e 2, sobre a estrutura dos subcorpos intermediários a uma extensão deste tipo, conhecimentos quais são obtidos pela atuação forte da Teoria de Galois, uma vez que é muito bem conhecida a estrutura de subgrupos do grupos dos Quatérnios. Finalmente, em posse dos resultados e caracterizações dos capítulos precedentes, juntamente aos resultados que relacionam formas quadráticas e álgebras quaterniônicas, no capítulo 6 demonstramos o Critério de Witt, que acerta sobre a imersão de extensões biquadráticas em quaterniônicas. Deste critério obtemos um importante resultado de interesse da Teoria dos Números, uma nova caracterização dos números racionais que são somas de três quadrados / Mestrado / Algebra / Mestre em Matemática

No description available.

Square Form Factoring is an <i>O</i>(<i>N</i>^1/4) factoring algorithm developed by D. Shanks using certain properties of quadratic forms. Central to the original algorithm is an iterative search for a square form. We propose a new subexponential-time algorithm called SQUFOF2, based on ideas of D. Shanks and R. de Vogelaire, which replaces the iterative search with a sieve, similar to the Quadratic Sieve.

An integral quadratic form is said to be strictly regular if it primitively represents all integers that are primitively represented by its genus. The goal of this dissertation is to extend the systematic investigation of the positive definite ternary primitive integral quadratic forms and lattices that are candidates for strict regularity. An integer that is primitively represented by a genus, but not by some specific form in that genus, is called a primitive exception for that form. So, the strictly regular forms are those forms for which there are no primitive exceptions. Our computations of primitive exceptions for each of the 119 positive definite regular ternary forms which lie in multiple-class genera, and of the companion forms in their genera, show that there are 45 inequivalent such forms that are candidates for strict regularity. We provide a proof of the strict regularity of one of these candidates, bringing the total number of forms for which such proofs are known to 15, and prove partial results on the integers primitively represented by the other form in its genus. The theory of primitive spinor exceptional integers is used to analyze the primitive exceptions for the forms in two other genera known to contain a regular ternary form. In these cases, results are obtained relating the primitive representation of certain integers c by a given form in one of these genera to the primitive representation of the integers 4c and 9c by the forms in the genus.

Thesis advisor: Benjamin V. Howard / Given a field <italic>F</italic>, an algebraic closure <italic>K</italic> and an <italic>F</italic>-vector space <italic>V</italic>, we can tensor the space <italic>V</italic> with the algebraic closure <italic>K</italic>. Two quadratic spaces of the same dimension become isomorphic when tensored with an algebraic closure. The failure of this isomorphism over <italic>F</italic> is measured by the Hasse invariant. This paper explains how the determinants and Hasse Invariants of quadratic forms are related to certain cohomology classes constructed from specific short exact sequences. In particular, the Hasse Invariant is defined as an element of the Brauer group. / Thesis (MA) — Boston College, 2010. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.