Introduction to Algebra
Code
10820
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Vitor Hugo Bento Dias Fernandes
Weekly hours
6
Total hours
56
Teaching language
Português
Objectives
The student is supposed to learn on fundamental aspects of groups and rings, as well as on Factorization Theory in rings (in particular, of polynomials).
Subject matter
I. Groups
1. Basics.
2. Subgroups.
3. Cyclic groups.
4. Cosets. Index of a subgroup.
5. Congruence relations. Quotient groups. Normal subgroups.
6. Morphisms.
7. Canonical decomposition and Homomorphism Theorem.
8. Isomorphism theorems.
9. Symmetric Group.
II. Rings
1. Basics.
2. Zero divisors. Integral domains. Division rings.
3. Characteristic of a ring.
4. Subrings.
5. Congruence relations. Quotient rings. Ideals.
6. Morphisms.
7. Canonical decomposition and Homomorphism Theorem.
8. Isomorphism theorems.
III. Theory of Factorization
1. Divisors.
2. Prime and coprime elements.
3. Gauss semigroups.
4. Gauss rings.
5. Principal ideal rings.
6. Euclidean domains.
IV. Rings of Polynomials
1. Rings of polynomials.
2. Division algorithm.
3. Polynomial functions.
4. Theory of factorization in rings of polynomials.
5. Irreducibility.
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 and 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
Classes consist of oral explanation of the theory and of solving problems.