The “infrastructure” of quadratic fields is a body of theory developed by Dan Shanks, Richard Mollin and others, in which they relate “reduced ideals” in the rings and sub-rings of integers in quadratic fields with periodicity in continued fraction expansions of quadratic numbers. In this thesis, we develop cubic analogs for several infrastructure theorems. We work in the field K=Q(), where 3=m for some square-free integer m, not congruent to ±1, modulo 9. First, we generalize the definition of a reduced ideal so that it applies to K, or to any number field. Then we show that K has only finitely many reduced ideals, and provide an algorithm for listing them. Next, we define a sequence based on the number alpha that is periodic and corresponds to the finite set of reduced principal ideals in K. Using this rudimentary infrastructure, we are able to establish results about fundamental units and reduced ideals for some classes of pure cubic fields. We also introduce an application to Diophantine approximation, in which we present a 2-dimensional analog of the Lagrange value of a badly approximable number, and calculate some examples.

Submitted by Eliton Mendonça Moro null (elitonmoro@hotmail.com) on 2017-02-07T16:23:35Z No. of bitstreams: 1 Dissert Moro E M.pdf: 1346120 bytes, checksum: fb365a8ed97b3769301b908d77114d7c (MD5) / Approved for entry into archive by LUIZA DE MENEZES ROMANETTO (luizamenezes@reitoria.unesp.br) on 2017-02-13T16:33:25Z (GMT) No. of bitstreams: 1 moro_em_me_sjrp.pdf: 1346120 bytes, checksum: fb365a8ed97b3769301b908d77114d7c (MD5) / Made available in DSpace on 2017-02-13T16:33:25Z (GMT). No. of bitstreams: 1 moro_em_me_sjrp.pdf: 1346120 bytes, checksum: fb365a8ed97b3769301b908d77114d7c (MD5) Previous issue date: 2017-01-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Os sistemas de comunicação com Múltiplas Entradas e Múltiplas Saídas (MIMO), são sistemas constituídos por estruturas que utilizam várias antenas, tanto no transmissor como no receptor. Por serem transmitidos via antenas, naturalmente surgem problemas de ruídos e de multipercursos, que impõe um desafio para o desenvolvimento dos sistemas de comunicação MIMO. Por esses motivos, muitos estudos focam em certas propriedades dos sinais enviados a fim de minimizar os efeitos sofridos na informação durante a transmissão. Existem muitos tipos diferentes de Códigos de Bloco Espaço-Temporais (STBC) disponíveis para duas antenas transmissoras, dentre eles, o código de bloco espaço-temporal ciclotômico, Código de Ouro e Código de Prata. Neste trabalho apresentamos uma construção de STBC cujos os sinais utilizados na transmissão são identificados por elementos de anéis de inteiros de corpos de números totalmente imaginários, Q(√d), com d<0, e apresentamos os melhores STBC em termos do critério que denominamos como critério produto, considerando extensões de Q(√d) com d=-1,-2,-3,-7, -11. / The communication systems of Multiple Input and Multiple Output (MIMO), are systems consisting of structures that use multiple antennas, both on the transmitter and the receiver. For being transmitted via antennas, noise and path problems naturally arise, which poses a challenge for the development and optimization of MIMO systems. For these reasons, many studies focus on certain properties of the signals sent in order to minimize the effects suffered on the information during transmission. There are many different types of Space-Time Block Codes (STBC) available for two transmitting antennas, such as the cyclotomic space-time block code, Golden code, and Silver code. In this work, we present a STBC construct via totally imaginary quadratic fields, Q(√d) with d <0 and present the best STBC in terms of the criterion that we call product criteria, considering extensions of Q(√d) with d = -1, - 2, - 3, - 7, -11.

Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known. Hooley derived a conjectural formula for the average of class numbers of real quadratic fields. In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment, i.e., the average of an arbitrary real power of class numbers. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley's work. In the final chapter we consider the case of the -1 power from a numerical point of view and develop an efficient algorithm to compute the average for the -1 class number power without computing class numbers.

Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known. Hooley derived a conjectural formula for the average of class numbers of real quadratic fields. In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment, i.e., the average of an arbitrary real power of class numbers. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley's work. In the final chapter we consider the case of the -1 power from a numerical point of view and develop an efficient algorithm to compute the average for the -1 class number power without computing class numbers.

Let Q(√(-d)) be an imaginary quadratic field with discriminant Δ. We use the isomorphism between the ideal class groups of the field and the equivalence classes of binary quadratic forms to find the structure of the class group. We determine the structure by combining two of Shanks' algorithms [7, 8]. We utilize this method to find fields with cyclic factors that have order a large power of 2, or fields with class groups of high 5-ranks or high 7-ranks. / Master of Science

Ce mémoire s'emploie à étudier les corps quadratiques réels ainsi qu'un élément particulier de tels corps quadratiques réels : l'unité fondamentale. Pour ce faire, le mémoire commence par présenter le plus clairement possible les connaissances sur différents sujets qui sont essentiels à la compréhension des calculs et des résultats de ma recherche. On introduit d'abord les corps quadratiques ainsi que l'anneau de ses entiers algébriques et on décrit ses unités. On parle ensuite des fractions continues puisqu'elles se retrouvent dans un algorithme de calcul de l'unité fondamentale. On traite ensuite des formes binaires quadratiques et de la formule du nombre de classes de Dirichlet, laquelle fait intervenir l'unité fondamentale en fonction d'autres variables. Une fois cette tâche accomplie, on présente nos calculs et nos résultats. Notre recherche concerne la répartition des unités fondamentales des corps quadratiques réels, la répartition des unités des corps quadratiques réels et les moments du logarithme de l'unité fondamentale. (Le logarithme de l'unité fondamentale est appelé le régulateur.) / This memoir aims to study real quadratic fields and a particular element of such real quadratic fields : the fundamental unit. To achieve this, the memoir begins by presenting as clearly as possible the state of knowledge on different subjects that are essential to understand the computations and results of my research. We first introduce quadratic fields and their rings of algebraic integers, and we describe their units. We then talk about continued fractions because they are present in an algorithm to compute the fundamental unit. Afterwards, we proceed with binary quadratic forms and Dirichlet's class number formula, which involves the fundamental unit as a function of other variables. Once the above tasks are done, we present our calculations and results. Our research concerns the distribution of fundamental units in real quadratic fields, the disbribution of units in real quadratic fields and the moments of the logarithm of the fundamental unit. (The logarithm of the fundamental unit is called the regulator.)

Ce mémoire s'emploie à étudier les corps quadratiques réels ainsi qu'un élément particulier de tels corps quadratiques réels : l'unité fondamentale. Pour ce faire, le mémoire commence par présenter le plus clairement possible les connaissances sur différents sujets qui sont essentiels à la compréhension des calculs et des résultats de ma recherche. On introduit d'abord les corps quadratiques ainsi que l'anneau de ses entiers algébriques et on décrit ses unités. On parle ensuite des fractions continues puisqu'elles se retrouvent dans un algorithme de calcul de l'unité fondamentale. On traite ensuite des formes binaires quadratiques et de la formule du nombre de classes de Dirichlet, laquelle fait intervenir l'unité fondamentale en fonction d'autres variables. Une fois cette tâche accomplie, on présente nos calculs et nos résultats. Notre recherche concerne la répartition des unités fondamentales des corps quadratiques réels, la répartition des unités des corps quadratiques réels et les moments du logarithme de l'unité fondamentale. (Le logarithme de l'unité fondamentale est appelé le régulateur.) / This memoir aims to study real quadratic fields and a particular element of such real quadratic fields : the fundamental unit. To achieve this, the memoir begins by presenting as clearly as possible the state of knowledge on different subjects that are essential to understand the computations and results of my research. We first introduce quadratic fields and their rings of algebraic integers, and we describe their units. We then talk about continued fractions because they are present in an algorithm to compute the fundamental unit. Afterwards, we proceed with binary quadratic forms and Dirichlet's class number formula, which involves the fundamental unit as a function of other variables. Once the above tasks are done, we present our calculations and results. Our research concerns the distribution of fundamental units in real quadratic fields, the disbribution of units in real quadratic fields and the moments of the logarithm of the fundamental unit. (The logarithm of the fundamental unit is called the regulator.)