In this course we present to the first year student of the degree course in Mathematics the basic mathematical language and a first introduction to abstract algebraic structures through the preliminary analysis of the algebraic structures of the set of integers and of the set of polynomials a coefficients in a field.
Provide basic mathematical language. Introduction to abstract algebraic notions by studying the algebra of integers, polynomials in a rational, real, complex or finite fileds coefficient variable and their quotients. First introduction to abstract group theory.
By attending and partecipating to the activities (lectures both theretical and exercises, guided exercises, tutoring sessions, etc.) the student will
- Become familiar with the basic mathematical language and improve formalization skills.
- Become familiar with algebraic structures, first through the study of already known objects (integers, polynomials), then with an abstraction towards more theoretical notions (groups, quotients, homomorphisms etc.).
Standard blackboard lectures with studentes and teachers in presence.
Introduction to the basic notions: sets, maps, surjectiive, , injective and bijective.
- Binary operation and their properties. Equivalence relations and induced quotients.
- Cardinality: countsble and uncountable sets. Permutations. Induction. Newton binomial formula.
- The integers: Euclidean algorithm and its applications. Prime numbers and unique factorization. The integers mod n.
- Complex numbers.
- Polinomials: in one variable and coefficients in the rational numbers, real numbers, complex numbers and finite fields. . Unique factorization property for polynomials. Irreducibility criteria. Quotients and their properties: zero-divisors, nilpotents and invertible elements.
- Introduction to algebraic structures. Groups, perionds, subgroups, homomorphisms and quotients.
Luca Barbieri-Viale, "Che cosa e' un numero?", Cortina Ed. 2013.
Lindsay N. Childs, "Algebra, un'introduzione concreta", (traduzione di Carlo Traverso), ETS Editrice Pisa, 1989.
M. Artin, Algebra, Bollati Boringhieri
I. N. Herstein, Algebra, Editori Riuniti
Teachers will also provide written notes in italian via aula-web
Ricevimento: By appointment
Ricevimento: To be decided later on when the general timetable will be fixed.
ALDO CONCA (President)
MARIA EVELINA ROSSI
ALESSANDRO DE STEFANI (President Substitute)
The class will start according to the academic calendar.
The exam consists of a written test and an oral exam.
Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the office Settore Servizi di supporto alla disabilità e agli studenti con DSA and the teachers at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools. Good afternoon !
