Teachings

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Professor
ANDREA LOI (Tit.)
JORGE NUNO DOS SANTOS VITORIA
Period
Second Semester 
Teaching style
Convenzionale 
Lingua Insegnamento
ITALIANO 



Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/64]  MATHEMATICS [64/00 - Ord. 2017]  PERCORSO COMUNE 10 80

Objectives

The aim of this course is to provide students with the basic elements of algebraic structures such as groups and rings, fundamental in a degree course in Mathematics.
The theoretical structure of the teaching consists in the development of the themes of the program, through the introduction of fundamental concepts and the development of a series of theorems with related demonstrations, supported by significant examples, exercises and applications.
In particular, the teaching aims to acquire a good knowledge of the theory of groups and rings. An important application will be the Frobenius-Stickelberger Theorem on the classification of finite abelian groups. The last part of the course will be dedicated to the ring of polynomial and the constructions with ruler and compass.

- APPLIED KNOWLEDGE AND UNDERSTANDING
At the end of the course the student must: be able to apply the concepts and techniques learned both to standard exercises and to the resolution of new problems, which require the independent elaboration of a strategy, or of small rigorous demonstrations, not identical to those already known but inspired by them.

- JUDGMENT AUTONOMY
Knowing how to recognize when a logical procedure is correct. Learn the standard demonstration techniques of abstract algebra.

- COMMUNICATION SKILLS
The student will be able to explain and solve problem solving; it will also be able to discuss and demonstrate correctly the most relevant results concerning basic algebra.

- ABILITY TO LEARN
Ability to learn how to solve complex exercises and problems independently. Ability to be able to read and understand an advanced text of mathematics.

Prerequisites

Injective and surjective maps; partially and totally ordered sets; Zorn Lemma; Equivalence and partition; prime numbers and the fundamental theorem of the algebra; Euler functions; linear and affine transformations; matrix calculation.

Contents

For a detailed programme see loi.unica.it

Algebraic structures; semigroups, monoids, groups, rings and fileds. Groups and subgroups: permutation groups, subgroups; cosets classes; normal subgroups, linear groups.
Homomorphisms and direct products: direct products of groups; quotient of grous, homomorphismi of groups; the theorems of homomorphisms for groups; the groups of automorphiisms of a group direct product of groups.

Abelian groups: cyclic groups; finite abelian groups, some infinite abelian groups.
Rings and ideals: definitions and examples; rules in a ring; the division ring of quaternions; subrings; ideals; quotient rings; prime and maximal idelas in commutative rings. Homomorphisms and direct product of rings: homomorphiisms and kernels; homomorphism theorem for rings. The field of quotients of an integral domain. The polynomial ring over a field. Irreducibility criteria. Algebraic extensions. Constructions with ruler and compass.

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 written test lasts 180 minutes and consists of four exercises (the first two on the theory of groups, the other two on the theory of rings, field extensions and constructions with ruler and compass). A positive outcome of the written test is required to gain access to the oral exam. The oral part of the examination (blackboard or teams) is about 45 minutes with questions on main parts of the course program. A negative outcome of the oral test requires the repetition of the entire procedure (written and oral in series ) . The final vote, out of thirty, is a weighted average between the result of the written and oral exam. The exams will be held in person if the Covid-19 health crisis allows for it. Alternatively, the exams will take place on the Teams platform or on an alternative telematic platform previously 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.

More Information

In http://people.unica.it/andrealoi/didattica/materiale-didattico/ students can find the detailed program and the exercises during class.

There is no student reception time. The student can apply for an appointment with the teacher 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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