60/64/184 - ALGEBRA 2
Academic Year 2019/2020
Free text for the University
ANDREA LOI (Tit.)
- Teaching style
- Lingua Insegnamento
|[60/64] MATHEMATICS||[64/00 - Ord. 2017] PERCORSO COMUNE||10||80|
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.
- 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.
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.
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 field of quatwernions; subrings; idelas; quotient rings; prime and maximal idelas in commutative rings.
Homomorphisms and direct product of rings: homomorphiisms and kernels; homomorphiisms theorem for rings; unitary rings and field of quotient of a domain, direct product of rings.
Blackboard and slides during the lectures, personal computer.
Verification of learning
The written test lasts 120 minutes and consists of two exercises
(the first on the theory of groups and the second on the theory of rings). A positive outcome of the written test (at least the complete solution of one of the exercise) is required to gain access to the oral exam .
The oral part of the examination 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 .
I.N.Herstein, Topics in Algebra, Second Edition.
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: