Approach to the algebraic theory of numbers

• Grant number: 14/02578-2
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: May 01, 2014
• End date: April 30, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Lucio Centrone
• Grantee: Guilherme Vicentin de Toledo
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

We intend with this scientific initiation that the student has anintroduction to algebraic number theory, the study of elements thatgeneralizes our notion of "integers" with algebraic structures. This studyis justified by the vanguard position that this discipline has taken insolving the classical problems of number theory and by its historicalpower. Starting with a brief study of groups, especially in the area ofrepresentations, rings and Galois theory, we will then focus on the classicquestions of algebraic number theory: factoring ideals, focusing onDedekind domains; theory of valuations, by means of Euclidean functions andby topological means; the search for units, invertible elements, andconcepts of how to circumvent its existence by the theory related to theclass number; and local fields, or the local completion of algebraicstructures, among other topics. We expect the student to end up with basicskills in the modern study of number theory, with the advantage of havingobtained a thorough knowledge in various branches of modern algebra. At theend of the initiation, the student must write a text to develop what hehas studied.