Algebra

Code:

M2032

Acronym:

M2032

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2022/2023 - 1S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:M	108	Official Study Plan	2	-	9	84	243

Teaching language

Portuguese

Objectives

To introduce the concepts, methods and basic results of Group and Ring Theory.

Learning outcomes and competences

Upon completing this curricular unit, the student should:

(1) master the basic concepts, methods, and results of Group and Ring Theory;

(2) to be able to analyze and solve problems within Group and Ring Theory, using the methods and results that best apply to the problems under study;

(3) to appreciate the connections of Group and Ring Theory to other areas of mathematics, such as Geometry and Number Theory;

(4) be able to efficiently and clearly communicate their resolutions of problems, and their understanding of the subject.

Working method

Presencial

Program

• Basic notions: binary, order and equivalence relations.


• Definitions and elementary properties of groups and rings; important examples of groups and rings: permutation groups; the integers, the real and the complex numbers; integers modulo n; permutations; matrices and linear groups; symmetry groups; polynomial rings and formal power series.


• Division rings, integral domains and fields. Examples.


• Direct product of groups and rings.


• Subgroups and subrings.


• Generators of a group; cyclic groups; cosets; Lagrange's theorem and some of its consequences, such as Fermat's “little” theorem and its generalization by Euler, as well as the fact that every prime-order group is cyclic.


• Homomorphisms and isomorphisms. Conjugation. Cayley's Theorem.


• Reference to the fundamental theorem of finitely generated abelian groups.


• Normal subgroups and quotient groups. Fundamental theorem on homomorphism for groups.


• Group actions and applications. Sylow's Theorems.


• Ideals and quotient rings.


• Fundamental Theorem on Homomorphism for rings.


• Finite fields.


• Reference to Euclidean domains, principal ideal domains, irreducible elements and prime elements in the context of integers and polynomials with coefficients in a field.

Mandatory literature

Gregory T. Lee; Abstract Algebra: an Introductory Course, Springer, 2018. ISBN: 978-3-319-77648-4

Complementary Bibliography

Fernandes Rui Loja; Introdução à álgebra. ISBN: 972-8469-27-6
Rotman Joseph; A first course in abstract algebra. ISBN: 0-13-011584-3
B. L. van der Waerden; A history of algebra from Al-Khwarizmi to Emmy Noether. ISBN: 3-540-13610-X
Fraleigh John B.; A first course in abstract algebra. ISBN: 0-201-16847-2
Peter M. Neumann; Groups and geometry. ISBN: 0-19-853451-5
M. A. Armstrong; Groups and symmetry. ISBN: 0-387-96675-7
J.S. Milne; Group Theory, 2017

Teaching methods and learning activities

The contact hours are distributed in theoretical and theoretical-practical classes. In the first ones, the contents of the program are studied, often using examples to illustrate the concepts treated and to guide the students in the resolution of exercises and problems. In the theoretical-practical classes, exercises and problems are solved, which are indicated in advance for each week. List of exercises and other course materials are available on the course page at Sigarra.

keywords

Physical sciences > Mathematics > Algebra > Group theory

Evaluation Type

Evaluation with final exam

Assessment Components

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

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	159,00
Frequência das aulas	84,00
Total:	243,00

Eligibility for exams

There are no rules concerning attendance frequency.

Calculation formula of final grade

Approval in the course unit may be obtained through two tests or  in the final exam.

The tests, to take place in dates to be announced, correspond to course material on Group Theory and Ring Theory, and will take respectively two hours and one hour and have respective weights of 2/3 and 1/3 in the computation of the course grade.

Students approved in the tests are exempt from participating in the final exam but may still participate in it. The exam will consist in two parts, corresponding to the same division of the course material, with the same duration and weights as the tests. Students may choose to take only one part of the exam if they so wish. The final grade is computed by taking for each part the maximum between grades obtained in the tests and in the exam.

The examination of the "appeal" period will be made in the same way as the one of "normal" period.

Special assessment (TE, DA, ...)

Examinations required under special statutes shall consist of a written test that may be preceded by an oral test, to assess if the student satisfies minimum conditions to attempt to obtain approval at the discipline in the written test.