Algebra II
Code
10981
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Vitor Hugo Bento Dias Fernandes
Weekly hours
5
Total hours
70
Teaching language
Português
Objectives
Theory of factorization in rings, rings of polynomials and field extensions. Brief introduction to Galois theory.
Prerequisites
None.
Subject matter
I. Theory of Factorization
1. Divisors.
2. Prime and coprime elements.
3. Gauss semigroups.
4. Gauss rings.
5. Principal ideal rings.
6. Euclidean domains.
II. Rings of Polynomials
1. Rings of polynomials.
2. Division algorithm.
3. Polynomial functions.
4. Theory of factorization in rings of polynomials.
5. Irreducibility.
III. Field extensions
1. Prime fields.
2. Extensions. Simple extensions. Algebraic extensions.
5. Algebraically algebraic closed fields and algebraic closure of a field.
6. Rupture and splitting fields.
7. Finite fields.
IV. Elements of Galois Theory
1. The Galois group.
2. Normal and separable extensions.
3. The Galois correspondence.
4. Solving equations by means of radicals.
Bibliography
1. J. Durbin, Modern Algebra, John Wiley & Sons, Inc.
2. N. Jacobson, Basic Algebra I, W. H. Freeman and Company.
3. S. Lang, Algebra, Addison-Wesley Publishing Company, Inc.
4. A. J. Monteiro e I. T. Matos, Álgebra, um primeiro curso, Escolar Editora.
5. M. Sobral, Álgebra, Universidade Aberta.
6. G.M.S. Gomes, Anéis e Corpos, uma introdução, DM-FCUL, 2011.
Teaching method
Lectures + problem-solving sessions (5h00).
Evaluation method
Students must attend, at least, 2/3 of classes.
Each student will be evaluated by 2 tests and at least 2 mini-tests or an exam. More details in the portuguese version.