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 2025
- Offered
- 2025/26
- Learning activities
- Caratterizzante
- Area
- Formazione matematica 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 primary aim of this course is to provide students with a solid foundation in algebraic structures, preparing them for more advanced studies in mathematics. Emphasis is placed on a deep understanding of theoretical arguments and on precision and rigor in the presentation of concepts and reasoning. Knowledge and understanding Mathematical understanding of the main topics related to groups, rings, and fields. Knowledge of the theoretical content covered in the course and of its key examples. Assessment methods: written and oral examination. Applying knowledge and understanding Ability to independently read and understand basic algebra textbooks. Ability to relate different theoretical topics and to identify examples and counterexamples. Ability to construct simple rigorous proofs and solve new problems that are clearly related to the topics covered. Assessment methods: written and oral examination. Making judgements The presentation of content aims to enhance the student's ability: to recognize valid proofs; to detect logical flaws or incorrect reasoning. Communication skills The course aims to develop: the ability to clearly and accurately communicate problems, ideas, and solutions in algebra; both in written and oral form; with appropriate use of mathematical language.
- Prerequisites
- To successfully follow the course, a solid understanding of the topics covered in a first course in Algebra is required. Specifically, students should be familiar with: the properties of natural and integer numbers, including Euclidean division and residue classes modulo an integer; fundamental properties of functions, especially invertibility; relations and finite/infinite cardinalities, including basic notions of combinatorics.
- 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.
The introduction of two intermediate written exam will be considered, in this case 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) two written tests that take place during the lessons. All written exams (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. Cauchy 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. - Obiettivi Agenda 2030 per lo sviluppo sostenibile