Algebra I
Objectives
The student is supposed to learn about fundamental aspects of groups and rings.
General characterization
Code
12910
Credits
6.0
Responsible teacher
Gonçalo Jorge Trigo Neri Tabuada
Hours
Weekly - 4
Total - Available soon
Teaching language
Português
Prerequisites
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.
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.
Teaching method
Lectures and problem-solving sessions (4h00).
Evaluation method
There are two mid-term tests. The final mark is the arithmetic mean of the non-rounded grades of the tests.
To be approved in final exam, the student must have a minimum grade of 9.5 in it.
More detailed rules are available in the portuguese version.
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.
Programs
Programs where the course is taught: