Fundamental Algebra

Code:

M501

Acronym:

M501

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	8	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

The focus of the course is on traditional algebra topics. Some of the topics to be covered are basic or simple, and familiar to some of the students, while others are more advanced.

Learning outcomes and competences

It is expected that students learn Group Theory, in particular Sylow theorems, ring theory, namely Unique Factorization Domains and Principal Ideal Domains, and also in Field Extensions and Galois Theory. It is also expected that students learn some more advanced topics in some of these areas.

Working method

Presencial

Program

Groups, Subgroups, Homomorphisms and the Isomorphism Theorems. Free groups and Presentations.

Group Actions, Sylow Theory. Small Groups. Composition Series. Nilpotent and Solvable Groups.

Rings. Subrings and Ideals. Homomorphisms. Domains and Fields. Polynomials. Principal Ideal Domains. Unique Factorization Domains. Noetherian Rings. Groebner Bases.

Field extensions. Algebraic Extensions, Separable, Purely Inseparable and Transcendental Extensions.

Galois theory. Splitting Fields. Normal Extensions. Galois Extensions. Solvability by Radicals. Geometric Constructions.

Depending on the background and interests of the students, some topics may be developed in considerably more depth than others.

Mandatory literature

Pierre Antoine Grillet; Abstract Algebra, Springer, 2007. ISBN: 978-0387715674

Complementary Bibliography

David S. Dummit and Richard M. Foote; Abstract Algebra, John Wiley & Sons, Inc., 2004. ISBN: 978-0471433347

Teaching methods and learning activities

Lectures: presentation of the course material and of examples by the teacher.
Office hours for student advice in time to be arranged individually.

Software

GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra

keywords

Physical sciences > Mathematics > Algebra

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Eligibility for exams

Course registration is the only requirement.

Calculation formula of final grade

The students' grades were based on the following activities:
75% homework assignments (there will be 4 problem sets; however, the lowest problem set score will be dropped);
25% Final exam.
The classification obtained through the homework assignements can not be used in the remaining exams.

Classification improvement

Grade improvement can be attempted only through examination.