Subject: Algebra

Scientific Area:

Mathematics

80 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

# Overall objectives:

1 - To promote learning of fundamental concepts of Set Theory, Order Relations, Group Theory and Ring Theory.
2 - The cognitive development of students, through theoretical demonstrations, as well as examples and counter-examples of the analyzed algebraic structures. It is also intended that students are able to investigate conditions where certain results are valid, and in other cases this does not happen.
3 - To promote analysis and discussion of problems.

# Syllabus:

1 - Maps; equivalence relations.
2 - Finite and infinite sets.
3 - Cardinal of a set.
4 - Order relations.
5 - Groupoids.
6 - Semigroups.
7 - Groups.
8 - Cyclic Groups.
9 - Congruence relations. Quotient groups. Invariant subgroups.
10 - Morphisms of groups. Canonical decomposition of a morphism. Homomorphism Theorem. Isomorphism theorems.
11 - Direct product of groups.
12 - Solvable groups.
13 - Rings.
14 - Zero divisors. Integral domains, division rings and fields.
15 - Characteristic of a ring.
16 - Subrings, subdomains of integral domains, division subrings.
17 - Subring generated by a subset.
18 - Congruence relations. Quotient rings. Ideals.
19 - Morphisms of rings. Canonical decomposition of a morphism. Homomorphism Theorem. Isomorphism theorems.
20 - Direct sum of rings.

# Literature/Sources:

António J. Monteiro, Isabel Teixeira Matos , 2001 , Álgebra-um primeiro curso , Escolar Editora
Serge Lang , 1984 , Algebra , Addison-Wesley Publishing Company
Allenby, R. B. J. T. , 2003 , Rings, fields and groups : an introduction to abstract algebra , Butterworth Heinemann
Rui Loja Fernandes, Manuel Ricou , 2014 , Introdução à álgebra , IST Press

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
Lectures. Discussion and solving problems, which promotes the active participation of students. Additionally students are still required to propose new problems. The evaluation is done through the realization of two tests (50%+50%).