Unit ALGEBRA II
- Course
- Mathematics
- Study-unit Code
- GP006035
- Curriculum
- In all curricula
- Teacher
- Giuliana Fatabbi
- Teachers
-
- Giuliana Fatabbi
- Marco Timpanella
- Hours
- 63 ore - Giuliana Fatabbi
- 10 ore - Marco Timpanella
- CFU
- 9
- Course Regulation
- Coorte 2022
- Offered
- 2022/23
- Learning activities
- Caratterizzante
- Area
- Formazione teorica
- Academic discipline
- MAT/02
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Groups, rings, fields.
- Reference texts
- Dikranjan-Lucido, Aritmetica e algebra, Liguori (2007)
Herstein, Topics in Algebra, Wiley (1975)
Any supplementary material available in Unistudium - Educational objectives
- The main goal of the teaching is to provide students with basic knowledge in the field of algebraic structures, in order to then be able to undertake subsequent studies. Particular attention is given to the understanding of the arguments and to the rigor in the presentation of concepts and argumentations.
Knowledge and comprehension
Mathematical comprehension of the proposed topics, knowledge of the theory developed on groups, rings fields, and fundamental examples discussed on thi topic
Methods for verifying knowledge
Written and oral exam
Capacity
Be able to independently read and understand basic Algebra texts.
Connect arguments, find examples and counterexamples
Be able to produce simple rigorous proofs of mathematical results and problem solving of simple problems related to what has been illustrated in class
Methods for verifying skills
Written and oral exam
Autonomy of judgment.
The display of the contents and arguments will be carried out in order to improve the ability of the student to recognize rigorous proofs
Communication skills.
The presentation of the topics will be organized to allow the acquisition of a good ability to communicate problems, ideas and solutions concerning Algebra, both in written and oral form. - Prerequisites
- Good knowledge of the subjects of Algebra I, independently of the passing of such exam. What is needed in particular is a good confidence with the properties of natural and integer numbers (including the Euclidean division) and of the residue classes, of the functions and their invertibility, of the relations and finite cardinalities (including basic notions of combinatorics) an infinite cardinalities.
- Teaching methods
- face to face lessons and tutoring by advanced students
"assisted teaching" (exercises done with the help of the teacher and/or a tutor) - Other information
- Upon request, both the written test and the oral examination are given in English
Usage of platform "uni-studium" - Learning verification modality
- The exam consists of a written test and an oral test. The written test verifies the ability to produce rigorous demonstrations of problems and statements related to the topics of the course. The oral exam verifies the ability to clearly and rigorously explain some of the course contents.
Exemption from the written test can be obtained by passing (with an average of at least 18/30 and a a grade of 15/30 in each test) three exemption tests that take place during the lessons. All written tests (including exemptions) last about two hours and consist in solving three problems that can also be small parts of theory and serve to check the level of understanding of the topics covered and the ability to connect them. The written test of each session contains three exercises, one on groups, one on rings and one on fields. The oral exam, lasting about 45-60 minutes, tends to confirm the level of understanding of the topics covered and of critical study and personal re-elaboration.
For information on support services for students with disabilities and / or SLD, visit the page http://www.unipg.it/disabilita-e-dsa. The teacher is in each
case available to personally evaluate, in specific cases, any compensatory measures and / or personalized paths in the case of students with disabilities and / or SLD. The teacher is also available to evaluate any personalized courses for working or non-attending students - Extended program
- Algebraic structures. Permutations. Omomorphisms. Direct products. Normality and conjugates. Cuachy theorem and Sylow theory.
Fundamental theorem of omomorphisms for groups and rings. Prime and maximal ideals.
Euclidean, principal and factorial rings. Characteristic of rings and fields. Polynomial rings.
Ring and field extensions.