General Algebra

Objectives

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

General characterization

11584

9.0

Responsible teacher

Herberto de Jesus da Silva

Weekly - 4

Total - 60

Inglês

Prerequisites

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

Bibliography

1. J. Durbin, Modern Algebra, John Wiley & Sons, Inc.
2. T. Hungerford, Algebra, Springer, 1980.
3. N. Jacobson, Basic Algebra I, W. H. Freeman and Company
4. S. Lang, Algebra, Addison-Wesley Publishing Company, Inc.
5. 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.

Programs

Programs where the course is taught: