# General Algebra

## Objectives

It is intended that students acquire knowledge on Galois Theory, Category Theory, Module Theory and Commutative Algebra.

## General characterization

##### Code

11584

##### Credits

9.0

##### Responsible teacher

Herberto de Jesus da Silva

##### Hours

Weekly - 4

Total - 60

##### Teaching language

Inglês

### Prerequisites

Elementary knowledge of Group Theory and Ring Theory customarily provided in a Mathematics degree.

### Bibliography

- J. Durbin, Modern Algebra, John Wiley & Sons, Inc.
- T. Hungerford, Algebra, Springer, 1980.
- N. Jacobson, Basic Algebra I, W. H. Freeman and Company
- S. Lang, Algebra, Addison-Wesley Publishing Company, Inc.
- A. J. Monteiro e I. T. Matos, Álgebra, um primeiro curso, Escolar Editora.

### Teaching method

Classes consist on an oral explanation of the theory which is illustrated by examples and the resolution of some exercises.

### Evaluation method

There are two mid-term tests. These tests can substitute the final exam if CT is, at least, 9.5. CT 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. Elements of Galois Theory: The Galois group; Normal and separable extensions; The Galois correspondence; Solving equations by means of radicals.

II. Elements of Category Theory: Definition and examples of categories; Functors and natural transformations; Equivalence of categories; Products and coproducts; The Hom functors; Representable functors.

III. Elementary theory of Modules: Modules and module homomorphisms; Submodules and quotient modules; Direct sum and product; Free modules; Finitely generated modules; Exact sequences; Tensor product of modules.

IV. Introduction to Commutative Algebra: Prime ideals and maximal ideals; NiIradical and Jacobson radical; Operations on ideals; Rings and modules of fractions; Primary decomposition.