Class Abstract Algebra

This course aims to provide a solid foundation in the fundamentals of Abstract Algebra, with a special focus on the study of algebraic structures such as rings, fields, groups, and elementary applications to graphs. These concepts form the theoretical foundation of numerous areas of mathematics and computer science, notably cryptography, code theory, the study of symmetries, and algebraic modeling of computational problems. This course plays a fundamental role in the basic training of students in Mathematics and Computer Science, providing indispensable tools for understanding more advanced disciplines such as computational algebra, algebraic geometry, computer theory, and information security. Furthermore, it encourages abstract and structured thinking, essential for solving complex problems in both academic and professional contexts.

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List of contents Numbers, Polynomials, and Factorization Arithmetic in Integers: Divisibility, Greatest Common Divisor, Euclid's Algorithm Polynomials with Coefficients in Integers, Rationals, and Finite Fields Factorization of Polynomials and Irreducibility Criterion Rings, Domains, and Fields Definition of Ring, Integrity Domain, and Field Euclidean, Principal, and Factorial Domains Field Extensions and Fraction Fields Ring Homomorphisms and Ideals Ring Homomorphisms and Isomorphisms Kernel and Image Ideals and Principal Ideals Quotient Rings Homomorphism Theorem Introduction to Group Theory (Emphasis on Matrix Groups) Groups, Subgroups, Inverse Elements Permutation Groups and Matrix Groups (GL(n), SL(n)) Order of a Group and an Element Generated Subgroups, Covalent Classes Lagrange's Theorem Group Homomorphisms and Quotient Groups Group Actions and Applications Introduction to Graphs and Algebraic Modeling Finite-Field Cryptography CAS

Active and participatory methodologies will be used, such as collaborative problem-solving, the use of computer algebra software (e.g., SageMath), and the integration of real-world computer science applications. The use of digital simulations will allow for practical and contextualized consolidation of the content, fostering students' autonomy and critical thinking.

Gonçalves, Adilson - Introdução à Álgebra. Edição: 6. Publicação: IMPA, 2017. Páginas: 192. ISBN: 978-85-244-0430-6. West, D. B. (2001). Introduction to graph theory (Vol. 2, pp. 1-512). Upper Saddle River: Prentice hall.