# 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. Semigroups and 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.

10. Sylow theorem

11. Cayley theorem

12. Schur theorem

II. Rings and fields

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

M. Artin, Algebra. New Jersey, Prentice Hall, 1991.

P. J. Cameron, Notes on algebraic structures, https://cameroncounts.files.wordpress.com/2013/11/algstr.pdf

J. Durbin, Modern Algebra, John Wiley & Sons, Inc.

N. Jacobson, Basic Algebra I, W. H. Freeman and Company.

S. Lang, Algebra, Addison-Wesley Publishing Company, Inc.

A. J. Monteiro and I. T. Matos, Álgebra, um primeiro curso, Escolar Editora.

M. Sobral, Álgebra, Universidade Aberta.

J. Rotman, An introduction to the theory of groups, Springer, 1991.

D. Dummite & R. Foote, Abstract algebra, John Wiley & Sons, 2004.

J. M. Howie, Fundamentals of semigroup theory, London Mathematical Society, 1996.

GAP tutorial. https://www.gap-system.org/Manuals/doc/tut/chap0.html

### 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. Semigroups and 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.

10. Sylow theorem

11. Cayley theorem

12. Schur theorem

II. Rings and fields

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: