Fundamental Algebra

Recommended Prerequisites

Basic background in Mathematics, including basic Linear Algebra and an introduction to Abstract Algebra, covering the notions of vector space, group, ring, and field.

Teaching Methods

Lectures have an expository character, being up to the professor the choice of the most appropriate way to do it and the degree of participation of the students. As an integral part of the learning process, it may be recommended or required the solution of exercises, course projects (possibly with a computational component) or oral presentations.

Learning Outcomes

The goal of this course is to provide a general knowledge of Abstract Algebra at an advanced level. It is intended that the students become familiar with the main basic techniques and results of this area of Mathematics, or at least attain enough familiarity with some of  them to be able to acquire others by themselves that may later prove to be useful.

Work Placement(s)

No

Syllabus

Group actions, Sylow theory. Nilpotent and solvable groups. Free groups and presentations. Lie groups and algebraic groups. Groups with operators. Rings and modules. Hermite, Smith and Jordan normal forms for matrices. Wedderburn theory. Linear representations of groups. Polynomial rings and factorization theory. Field extensions. Galois theory. Norms, traces and discriminants. Ideal theory in commutative rings. Rings of integers. Dedekind domains. Algebraic sets and Hilbert's Nullstellensatz.

The program will cover most of the above topics. Depending on the background and interests of the students, some topics may be developed in considerably more depth than others.

Head Lecturer(s)

Manuel Augusto Fernandes Delgado

Assessment Methods

Assessment
Resolution Problems: 100.0%

Bibliography

D.S. Dummit and R.M. Foote, Abstract Algebra, 3rd edition, John Wiley & Sons, Inc., Hoboken, NJ, 2004.

P.A. Grillet, Abstract Algebra, 2nd edition, Graduate Texts in Mathematics, vol. 242, Springer, New York, 2007.

I.M. Isaacs, Algebra: a Graduate Course, Graduate Studies in Mathematics, vol. 100, AMS, 1994.