CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Este trabalho tem como objetivo mostrar que o conjunto dos nÃmeros reais à um corpo ordenado completo e que, a menos de um isomorfismo, à Ãnico. Este trabalho à voltado para todos aqueles que tenham interesse em MatemÃtica, sobretudo, para os professores de MatemÃtica do ensino mÃdio que utilizam as propriedades do conjunto dos nÃmeros reais sem conhecer a teoria matemÃtica envolvida. Para tanto, à necessÃrio caracterizar o conjunto dos reais a fim de provar suas propriedades. Aqui, utilizamos duas construÃÃes, a saber: os reais via sequÃncias de Cauchy devido a Cantor e os reais via Cortes de Dedekind. A partir dessas caracterizaÃÃes, conseguimos construir um corpo K munido das operaÃÃes de soma e multiplicaÃÃo onde mostramos que ele cumpre as condiÃÃes da definiÃÃo de corpo. Definida uma relaÃÃo de ordem em K, mostramos que tal corpo à ordenado e, alÃm disso, conseguimos mostrar que todo subconjunto de K admite supremo, o que quer dizer que tal corpo à completo. Finalmente, mostramos que qualquer outro corpo ordenado completo que possa, por ventura, existir à uma mera caracterizaÃÃo de ℝ, o que quer dizer que ℝ à Ãnico, a menos dessas possÃveis outras caracterizaÃÃes. Tal caracterizaÃÃo serà chamada de isomorfismo que à uma funÃÃo bijetora de ℝ para K. / This work aims to show that the set of real numbers is a complete ordered field that, within an isomorphism, is unique. This work is aimed at all those who are interested in mathematics, especially for that high school math teacher who uses the real numbers of the set of properties without knowing the mathematical theory involved. Therefore, it is necessary to characterize the set of the real in order to prove their properties. Here, we use two buildings, namely: the real via Cauchy sequences due to Cantor and the real via Dedekind cuts. From these characterizations, we can build a field K equipped with the addition and multiplication operations which show that it meets the definition of field conditions. Set an order relation in K, we show that such a body is ordered and in addition, we show that every subset of K admits supreme, which means that such a field is complete. Finally, we show that any complete ordered field that can, perchance appear is a mere characterization of ℝ, which means that ℝ is unique, unless these possible other characterizations. This characterization will be called isomorphism which is a function bijetora of ℝ to K.

La notion de modules de Drinfeld est le centre de cette thèse, cette notion fut introduite par Drinfeld en 1973, comme étant des " modules elliptiques" appelés de nos jours modules de Drinfeld. Ceux sont des objets algèbriques analogues aux courbes elliptiques sur les corps des nombres et sur les corps finis, obtenus par la réduction modulo une place non-archimédiennene. Une étude de l'arithmétique de tels objet devient légitime, motivée par l'arithmétique des courbes définies sur un corps fini initiée par Artin, Hasse et Weil. Dans cette direction on pousse cette analogie, pour un module de Drinfeld de rang 2, à la majorité de points étudiés pour des courbes elliptiques sur un corps fini. On donne plus précisement un analogue du théorème de Weil, théorème de Deuring-Waterhouse, et un analogue du travail de S. Vladut sur la cyclicité de tel structure algébrique.

Στο κεφάλαιο 1 γίνεται μελέτη διατεταγμένων αλγεβρικών δομών. Δίνονται ορισμοί, αποδείξεις και στοιχειώδη αποτελέσματα, απαραίτητα σε όλη την πορεία της εργασίας. Ορίζουμε μερικώς διατεταγμένα σύνολα και μερική διάταξη σε αλγεβρικά συστήματα, βλέπουμε υπό ποίες προϋποθέσεις η μερική διάταξη επεκτείνεται σε ολική και άρα το σύνολο γίνεται ολικώς διατεταγμένο και στη συνέχεια τα διατεταγμένα σύνολα με μία εσωτερική πράξη ορίζουν μερικώς ή ολικώς διατεταγμένες ομάδες. Στο κεφάλαιο 2 παρουσιάζουμε συμπληρώσεις διατεταγμένων συνόλων και συγκεκριμένα, τα συμπληρώματα Dedekind, Kurepa και Krasner καθώς και ορισμένες ιδιότητες αυτών. Ο Dedekind (1831-1916) όρισε τις τομές Dedekind με τη βοήθεια των οποίων επέκτεινε τη διάταξη των φυσικών στο σύνολο των πραγματικών και θεμελίωσε με αυτόν τον τρόπο το σύνολο αυτό ως ένα διατεταγμένο σώμα. Η κατασκευή της δομής των πραγματικών εφοδιασμένη με τις πράξεις της πρόσθεσης και του πολλαπλασιασμού και τη δίαταξη, καθώς και η κατασκευή της δομής του επιπέδου με τις ίδιες πραξεις και διάταξη κατά Dedekind παρουσιάζεται εκτενέστερα στο κεφάλαιο 3. Η γενίκευση της έννοιας του συμπληρώματος Kurepa και η εισαγωγή του συμπληρώματος Krasner, οφείλονται στον καθηγητή Λ. Ντόκα (1963). Η μέθοδος του Dedekind της συμπλήρωσης με τομές δεν είναι η μόνη μέθοδος κατασκευής των πραγματικών αριθμών. Η μέθοδος του G. Cantor (1845-1918) της συμπλήρωσης με ακολουθίες, είναι η δεύτερη εξίσου σημαντική μέθοδος, την οποία θα παρουσιάσουμε στο κεφάλαιο 4. Η μελέτη μας ολοκληρώνεται στο κεφάλαιο 5, όπου παρουσιάζεται ένα ενδιαφέρον αποτέλεσμα για τις μερικώς διατεταγμένες ομάδες και τις συνθήκες κάτω από τις οποίες αυτές επεκτείνονται σε ολικώς διατεταγμένες ομάδες, στηριζόμενοι στην εργασία “embedding groups into linear or lattice structures” των Κοντολάτου-Σταμπάκη (1987), όπου πραγματοποιούν επέκταση μίας μερικώς διατεταγμένης ομάδας, χρησιμοποιώντας τα αποτελέσματα του Fuchs για ύπαρξη επέκτασης ενός μερικώς διατεταγμένου συνόλου σε ολικώς διατεταγμένο. / In chapter 1 ordered algebraic structures are considered and we present certain definitions, proofs and elementary results which are necessary in the whole project. Partially ordered sets and partial order in algebraic systems is defined. Then we analyze under which conditions partial order can be extended to full order. This leads to fully ordered sets and those sets, along with an internal operation, define partially or fully ordered groups. In chapter 2 we present specific ordered set complements and in particular those of Dedekind, Kurepa and Krasner and furthermore we mention some of their properties. Dedekind sections where introduced by Dedekind (1831-1916), who used them in order to extend the order of natural numbers to the set of real numbers, making this set an ordered field. The construction of the real numbers structure along with the internal operations of addition and multiplication and order and the construction of the plane structure with the same operations and order, using Dedekind theory, is analytically presented in chapter 3. Due to L. Docas (1963), Kurepa complement was generalized and Krasner complement was introduced. Dedekind’s sections is not the only way to construct the set of real numbers. Another important method is that of G. Cantor (1845-1918), who used sequences for completion. We present this method in chapter 4. Finally, in chapter 5, we consider a paper published by A. Kontolatou and J. Stabakis (1987) entitled “Embedding groups into linear or lattice structures”. Fuchs’s results on the extend existence of a partially ordered set to fully ordered set is used. Based on the Kontolatou-Stabakis paper, we present an interesting result for partially ordered groups and certain conditions of how to extend those groups to fully ordered ones.

O teorema de Kronecker-Weber afirma que se K é uma extensão finita e galoisiana dos racionais com grupo de Galois abeliano, K tem que ser ciclotômica. / The Kronecker-Weber theorem stablishes that, if K is a Galois finite extension of Q with Galois group abelian, then K is a ciclotomic field.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Includes bibliographical references. / by Ilan Vardi. / Ph.D.

Made available in DSpace on 2015-06-17T19:34:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-12-13. Added 1 bitstream(s) on 2015-06-18T12:48:43Z : No. of bitstreams: 1 000832383.pdf: 314748 bytes, checksum: 79efa23654535c68b9a0c850fa367c1b (MD5) / Este trabalho busca formalizar a existência do conjunto dos números reais como um corpo ordenado completo. Mostraremos a construção de R a partir dos cortes de Dedekind e apresentaremos as principais propriedades deste conjunto. Apontaremos também as ideias básicas da construção de R através das Sequências de Cauchy, proposta por Cantor, e outras formas equivalentes. E, por fim, exemplificaremos como este conceito poderá ser aplicado em sala de aula pelos professores da Educação Básica / The aim of this work is to formalize the real numbers set existence as a complete ordered field. We will construct R using the Dedekind cuts and show the main properties of this set. Besides, we will point out the basic ideas of the Cantor's construction by Cauchy Sequences and another equivalent ways. The last part, consists of an example of how teachers can apply this concept in middle education classes

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Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-05-27T12:50:34Z No. of bitstreams: 1 arquivototal.pdf: 643760 bytes, checksum: c6fc649a3682bb07bcc815ff2163eef4 (MD5) / Approved for entry into archive by Leonardo Americo (leonardo@sti.ufpb.br) on 2015-05-27T12:52:35Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 643760 bytes, checksum: c6fc649a3682bb07bcc815ff2163eef4 (MD5) / Made available in DSpace on 2015-05-27T12:52:35Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 643760 bytes, checksum: c6fc649a3682bb07bcc815ff2163eef4 (MD5) Previous issue date: 2014-08-29 / This work aims to show the existence and Uniqueness of the field of Real Numbers, using for this, Dedekind' Cuts theorem and the Definition by Recursion.To fulfill his goal, we define the notion of Dedekind Cut and present some of its properties; then introduce the notions of Archimedean Ordered and Field, Complete Field Sorted and finally articulate and demonstrate the Uniqueness Theorem of Field Real Numbers. / Este trabalho tem como objetivo mostrar a Existência e a Unicidade do Corpo dos Números Reais, usando para isso, os Cortes de Dedekind e o teorema da defi- nição por Recursão. Para cumprirmos tal objetivo, definimos a noção de Corte de Dedekind e apresentamos algumas de suas propriedades; em seguida, apresentamos as noções de Corpo, Corpo Ordenado e Arquimediano, Corpo Ordenado Completo e, finalmente, enunciamos e demonstramos o Teorema da Unicidade do Corpo dos Números Reais.

Submitted by Jean Medeiros (jeanletras@uepb.edu.br) on 2017-09-12T12:42:53Z No. of bitstreams: 1 PDF - José Elias da Silva.pdf: 9535482 bytes, checksum: 4f018a51c1e15736072db257c4b86319 (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2017-10-26T16:05:13Z (GMT) No. of bitstreams: 1 PDF - José Elias da Silva.pdf: 9535482 bytes, checksum: 4f018a51c1e15736072db257c4b86319 (MD5) / Made available in DSpace on 2017-10-26T16:05:13Z (GMT). No. of bitstreams: 1 PDF - José Elias da Silva.pdf: 9535482 bytes, checksum: 4f018a51c1e15736072db257c4b86319 (MD5) Previous issue date: 2016-10-28 / In this study we work two constructions of the real numbers system. The construction the system of real numbers by cuts or straight sections in the set of rational numbers, advanced by Dedekind, and the construction of the real number as equivalence class of fundamental sequences of rational numbers, idea introducel by Cantor. Related to this approach, we dedicate a Chapter to show density of the rational num- bers and irrational numbers in the set of real numbers. Later, in a more synthesized form than the above constructions,we present other ap- proachs which the fundamental idea of real numbers is more clear. Finally we use method axiomatic in order to show the uniqueness of the real numbers system, thus, we conclude that there is a complete and orderly body which is unique up to isomorphism . This unique body is named the real numbers body. / Neste trabalho serão estudadas duas construções do sistema dos números reais. A construção do sistema dos números reais por cortes na reta ou secções no conjunto dos números racionais, avançada por Dedekind, e a construção do número real como classe de equivalência de sucessões fundamentais de números racionais, ideia protagonizada por Cantor. Relacionado com este tema, um capítulo deste trabalho será dedicado à aplicação da densidade dos números racionais e irracionais. Posteriormente, e de uma forma mais sintetizada do que nas anteriores, são apresentadas outras construções, procurando tornar mais claro a ideia fundamental subjacente ao conceito de número real. Por fim, utiliza-se o método axiomático com o intuito de mostrar a unicidade do sistema dos números reais, isto é, concluir finalmente que existe um corpo completo e ordenado, e apenas um a menos de um isomorfismo, do conjunto dos números reais.

Este trabalho apresenta a construção dos conjuntos dos números naturais, inteiros, racionais e reais, buscando contemplar uma mediação entre alunos e professores do ensino médio que possa contribuir em uma abordagem facilitadora para o processo de ensino e aprendizagem. A construção dos conjuntos numéricos é feita de modo progressivo, apresentando leis e propriedades que definem cada um deles. Os capítulos apresentam teoremas que são provados de modo que o leitor possa conseguir, efetivamente, estabelecer um elo entre a teoria matemática e suas abstrações iniciais inerentes aos estudantes em formação. / This work presents the construction of the sets of natural, integer, rational and real numbers, aiming to contemplate a mediation between high school students and teachers that can contribute to an easy approach to the teaching and learning processes. The construction of the numerical sets is done progressively presenting laws and properties that define each one of them. The chapters present theorems that are proven so that the reader can effectively establish a link between mathematical theory and its initial abstractions inherent in the students in formation.