Anéis de inteiros de corpos de números e aplicações

• Autor(a) principal: Araujo, Robson Ricardo de [UNESP]
• Data de Publicação: 2015
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: Repositório Institucional da UNESP
• Texto Completo: http://hdl.handle.net/11449/127767; http://www.athena.biblioteca.unesp.br/exlibris/bd/cathedra/14-09-2015/000846496.pdf

This master thesis presents the rings of integers of quadratic elds, cyclotomic elds, some cyclotomic sub elds and abelian number elds aiming use them to produce algebraic lattices, which are applied in the Information Theory and in the Error Correcting Codes Theory. The text develops basic concepts about Algebra and Algebraic Number Theory, studies integral basis of number elds from two di erent perspectives, characterizes the ring of integers of the aforementioned number elds and presents some applications of this theory to algebraic lattices. The main proven theorems in this thesis are Hilbert-Speiser Theorem and Leopoldt-Lettl Theorem. The second provides the ring of integers of any abelian number eld, generalizing the rst. This thesis has a chapter dedicated to make the proof of the Leopoldt-Lettl Theorem in detail. Furthermore, this work analyses the monogenesis of some ring of integers and presents a counterexample of a ring of integers non-monogenic. The last chapter is aimed at lattices and shows examples of algebraic lattices in spaces of dimensions 2, 4, 6 and 8 constructed by ideals of ring of integers of number elds through Minkowski homomorphism. The work that created this thesis consisted mainly in research and detailing of the proofs of Leopoldt- Lettl Theorem and of three theorems linked to the issue of monogenesis of the ring of integers. This e ort created a development lighter and less compact of the proofs related to these subjects, which is presented in the text. Finally, this thesis gathers and provides a great theoretical apparatus that has been useful to development of the theory of algebraic lattices and that creates the expectation of its use in future applications in this area