Teachings

Select Academic Year:     2016/2017 2017/2018 2018/2019 2019/2020 2020/2021 2021/2022
Professor
JORGE NUNO DOS SANTOS VITORIA (Tit.)
Period
Second Semester 
Teaching style
Convenzionale 
Lingua Insegnamento
ITALIANO 



Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/65]  MATHEMATICS [65/40 - Ord. 2020]  MATEMATICA PURA 9 72
[60/65]  MATHEMATICS [65/50 - Ord. 2020]  MATEMATICA PER LA DIDATTICA E LA DIVULGAZIONE 6 48

Objectives

KNOWLEDGE TRANSFER AND CONTENT COMPREHENSION
The aim of this course is to provide students with the opportunity to deepen their knowledge on classical topics of abstract algebra, through the study of Galois theory and an introduction to representation theory.

SCIENTIFIC SKILLS: The student ought to be acquire the capacity of analysing, understanding and applying the techniques necessary for the study of the algebraic objects under study.

AUTONOMY: The course aims to stimulate the autonomous use of all resources available to the students, from the contents taught in class, to the exercises proposed and the bibliography suggested.

COMMUNICATION SKILLS: The student should acquire the ability to communicate the contents of the course using adequate mathematical language, showing an appropriate degree of confidence in the use of algebraic structures and in the methods of proof.

LEARNING SKILLS: The student should develop an appropriate method of study, stimulating interpretation and analysis skills that are valuable both in future studies and in professional environments.

EXPECTED COMPETENCES: Development of the ability to communicate efficiently and correctly mathematical contents, through the use of a correct language and an organised and methodic presentation. This serves not only to pass this exam but also as preparation for future exams.

Prerequisites

Algebra 1, Algebra 2 and Geometry 1 (as offered at UniCa). This amounts to a good knowledge of linear algebra and a first course in abstract algebra (groups and rings). It is also required a basic knowledge of elementary logic and set theory.

Contents

I. Preliminaries
1. The language of categories. Examples
2. Some aspects of Group Theory: group actions, Sylow's first theorem and solvable groups.

II. Galois Theory
1. Splitting fields.
2. Finite fields.
3. Normal and separable extensions.
4. Intermediate fields and the Galois group.
5. Fundamental theorem of Galois theory.
6. Applications of Galois Theory

III. Introduction to representation theory
1. Algebras and modules. Examples.
2. Semisimplicity.
3. Maschke's theorem and Artin-Wedderburn's theorem.

Teaching Methods

If the Covid-19 health crisis allows, lectures will take place in person through the presentation of contents in the blackboard (or with the use of a tablet, with projection in the classroom). In case this is not possible, the lectures will take place in a mixed environment, with live online transmissions from the classroom.

Verification of learning

The course will be assessed through a weighted average between a written and an oral exam. To access the oral exam, the student must pass the written (mark greater or equal than 18 out of 30). If the student does not pass the oral exam, they will have to repeat the entire procedure (written and oral, in sequence). The exams will take place in person if the conditions of the Covid-19 health crisis allow for it. Otherwise, the exams will take place online, through Teams or through another platform to be agreed between the lecturer and the student.

Texts

I. N. Herstein, Topics in Algebra (2nd edition), 1975.
C. Pinter, A book of abstract Algebra, McGraw-Hill Book Company.
R. Pierce, Associative Algebras, Springer (1982).
I. Assem, D. Simson, A. Skowronski, Elements of the representation theory of associative algebras 1: Techniques of representation theory, London Mathematical Society Student Texts 65 (2006)
F.W. Anderson, K.R. Fuller, Rings and categories of modules (2nd edition), Springer 1992

More Information

In
https://www.unica.it/unica/it/ateneo_s07_ss01_sss03.page?contentId=SHD147070

students can find the exercises proposed in class and any additional materials suggested by the lecturer.

The office hours for the course will be announced on the website of the lecturer at the start of the course, and they will be announced in class.
The student may, in any case, ask for an appointment with the lecturer via e-mail.

Our University provides support for students with specific learning disability (SLD). Those interested can find more informations at this link:
http://corsi.unica.it/matematica/info-dsa/

Questionnaire and social

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