https://ri.ufs.br/jspui/handle/riufs/2568 Sun, 03 May 2026 18:50:41 GMT 2026-05-03T18:50:41Z https://ri.ufs.br/jspui/handle/riufs/24859 Título: O número de classes de um corpo de números Autor(es): Silva, Rafael Fagundes Bitencourt Abstract: This monograph aims to address the structure of number fields and their ring of integers, focusing on the class group. More specifically, it aims to study its order, proceeding from the algebraic basis to analytical constructions. As applications of the theory, we study the class group of two types of fields, which are essential to Algebraic Number Theory: the cyclotomic fields and quadratic fields. Wed, 04 Mar 2026 00:00:00 GMT https://ri.ufs.br/jspui/handle/riufs/24859 2026-03-04T00:00:00Z https://ri.ufs.br/jspui/handle/riufs/24841 Título: Sizígias de um Módulo Autor(es): Lopes, Igor de Santana Abstract: This work aims to analyze the syzygies of a module over a commutative ring, emphasizing the structure and properties of the syzygies module. Starting from the definition and characterization of nontrivial linear relations between generators, we demonstrate how these structures allow us to measure the distance of a module from the condition of being free. The research includes the study of the properties and behavior of syzygies, as well as their implications for understanding modular structures in the context of commutative algebra. Examples are presented to illustrate and substantiate the theoretical results, thus providing a clear approach to the topic. Fri, 13 Feb 2026 00:00:00 GMT https://ri.ufs.br/jspui/handle/riufs/24841 2026-02-13T00:00:00Z https://ri.ufs.br/jspui/handle/riufs/24840 Título: Potências transcendentes : uma jornada pelo Teorema de Gelfond-Schneider Autor(es): Santos, Mirelly Larissa Ferreira Abstract: This monograph focuses on the study of the Gelfond-Schneider Theorem, one of the most remarkable results in the theory of transcendental numbers, whose proof solved Hilbert’s seventh problem. The text guides the reader through a gradual journey of fundamental concepts, ranging from the notions of algebraic and transcendental numbers to classical results, such as Liouville’s Theorem, the transcendence of notable constants (such as e and π), and the Hermite-Lindemann Theorem. Then, it presents the methods and auxiliary results that underpin the proof by Gelfond and Schneider, also exploring consequences and applications of the theorem, such as the characterization of the transcendence of various algebraic powers. Thus, this work aims not only to present the proof of this result but also to contextualize its importance for the consolidation of transcendence theory and for solving open problems in modern mathematics. Wed, 04 Mar 2026 00:00:00 GMT https://ri.ufs.br/jspui/handle/riufs/24840 2026-03-04T00:00:00Z https://ri.ufs.br/jspui/handle/riufs/24028 Título: Uma introdução à base de Gröbner Autor(es): Pimentel, José Antônio Santos Abstract: In this work, we will develop a study in the area of Commutative Algebra, focusing on results related to a special topic known as the Gröbner Basis. The main subjects addressed include polynomial rings in several variables, highlighting how we can establish a monomial order and the division algorithm in the ring K[x1, . . . , xn]. The knowledge acquired throughout the first chapter will serve as the foundation for the study of this basis. In particular, we will analyze the importance and application of Buchberger’s algorithm, used to determine a Gröbner Basis. Finally, we will seek to apply the developed theory to the resolution of nonlinear systems, addressing practical examples such as the map coloring problem and road network systems. Tue, 07 Oct 2025 00:00:00 GMT https://ri.ufs.br/jspui/handle/riufs/24028 2025-10-07T00:00:00Z