OVERVIEW

In Algebra 2, we present the main abstract algebraic structures that were introduced informally and primarily through examples in Algebra 1. In particular, we discuss the concepts of groups and rings. Furthermore, we begin the study of field extensions. The lectures are conducted in Italian.

AIMS AND CONTENT

LEARNING OUTCOMES

The main concepts of abstract algebra that were introduced in a less formal way in Algebra 1 are presented. In particular, the notions and main properties of the algebraic structures of group and ring are discussed and  extensions of fields will be presented.

AIMS AND LEARNING OUTCOMES

Aims of the Course:

1. To introduce the fundamental algebraic concepts and the relationships between them.

2. To describe the construction of abstract algebraic objects, their representation, identification, and manipulation.

3. To provide a detailed and in-depth analysis of the following topics: free groups, group actions on sets and the associated decomposition into orbits, the classification of finitely generated Abelian groups, the isomorphism theorems, unique factorisation in rings, finite fields, algebraic elements and their minimal polynomials.

Learning Outcomes:

By the end of the course, students will be able to:

1. Understand various types of abstract algebraic structures and recognise their differences and similarities.

2. Determine whether a given abstract algebraic structure possesses certain properties.

3. Construct abstract algebraic structures with specified characteristics.

4. Reproduce and generalise constructions and theoretical arguments aimed at understanding and analysing abstract algebraic structures.

PREREQUISITES

The course requires familiarity with the concepts and examples of algebraic structures introduced in the Algebra 1 and Linear Algebra courses.

TEACHING METHODS

Teaching Methods:
The course is delivered through lectures given by the instructors, during which theoretical concepts will be presented, applied to concrete examples, and illustrated through the solution of exercises.

Students with certified Specific Learning Disorders (SLD), disabilities, or other special educational needs are encouraged to contact the lecturer and the School/Department disability coordinator at the beginning of the course to agree on appropriate teaching arrangements. These arrangements will respect the learning objectives while taking individual learning styles into account.

Students who are unable to attend lectures are invited to contact the lecturer to arrange dedicated meetings and access to specific course materials.

SYLLABUS/CONTENT

Course Content:

• Groups: Cyclic groups, subgroups, normal subgroups, and quotient groups. Homomorphisms and isomorphisms. Automorphism groups. Order of an element, Cartesian products of groups, semidirect products of groups. Group actions on sets, orbits and stabilisers. Linear groups, permutation groups, free groups, and group presentations. Finite groups of small order. Abelian groups, finitely generated Abelian groups, torsion groups. The structure theorem for finitely generated Abelian groups and its applications.

• Rings: Commutative rings and rings with unity. Units, zero divisors, nilpotent and idempotent elements. Rings that are fields. skew fileds, integral domains, reduced rings. Subrings and ideals. Quotient rings, maximal ideals, prime ideals, and radical ideals. The quaternions. Characteristic of a unital ring. Euclidean rings, Gaussian integers, principal ideal domains (PIDs), and unique factorisation domains (UFDs). Polynomial rings, evaluation homomorphisms, Noetherian rings. Canonical isomorphisms. The Chinese Remainder Theorem in a PID. Frobenius homomorphism. Finite fields. Field extensions. Algebraic and transcendental elements. Finite extensions. The splitting field of a polynomial.

RECOMMENDED READING/BIBLIOGRAPHY

Study Materials:
In general, lecture notes and the materials provided on the aul@web platform are sufficient for studying and preparing for both the written and oral examinations. The following books are recommended as supplementary resources for further study or for students who are unable to attend lectures:

• M. Artin, Algebra, Bollati Boringhieri

• Lindsay N. Childs, Algebra: A Concrete Introduction, ETS Editrice Pisa, 1989

• D. Dikranjan, M. Lucido, Aritmetica ed Algebra, Liguori Eds.

• Esercizi scelti di Algebra, Volume 1, R. Chirivì, I. Del Corso, R. Dvornicich, Springer Verlag, Unitext Series

• Esercizi scelti di Algebra, Volume 2, R. Chirivì, I. Del Corso, R. Dvornicich, Springer Verlag, Unitext Series

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

22 September 2025

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Written and oral

ASSESSMENT METHODS

If each of the intermediates written exams has been overcome with a grade >=16/30, and the average grade is at least 18/30, it is not necessary to take the written exam.

The written exam consists in exercises related to concepts seen during the lectures. Similar exercises are often done during the exercise-lectures.

The oral exams will concern concepts seen during the lectures, with the purpose to verify if the student has gained the necessary knowledge

FURTHER INFORMATION

Intermediete written tests.