The aim of the course is to introduce some of the basic notions of algebra. The students are expected to obtain a good understanding (both theoretical and practical) of some fundamental algebraic structures: groups and rings.
Course Prerequisites
The contents of the course of Linear Algebra.
Teaching Methods
Lectures and exercise sessions
Assessment Methods
The exam consists of a written test, during which the student must solve some exercises, and of an oral examination, during which the student must answer some questions, mainly of a theoretical nature. The written and oral tests must be taken in the same exam session.
Texts
Notes provided by the teachers. I.N. Herstein: "Algebra", Editori Riuniti. M. Artin: "Algebra", Bollati Boringhieri.
Contents
The integers. Integer division. Greatest common divisor and the Euclidean algorithm. Unique factorization of integers. Congruences. Groups: definition and examples; abelian groups. Subgroups. Homomorphisms and isomorphisms of groups; kernel of a homomorphism. Direct product of groups. Cyclic groups and generators of a group. Order of an element. Index of a subgroup and Lagrange's theorem. Normal subgroups; quotient group modulo a normal subgroup. Symmetric groups and Cayley's theorem. Homomorphism and isomorphism theorems for groups. Rings (commutative and non-commutative), integral domains, division rings and fields. Homomorphisms of rings. Ideals and operations on ideals. Quotient ring modulo a two-sided ideal. Homomorphism and isomorphism theorems for rings. Chinese remainder theorem. Prime and maximal ideals. Polynomials with coefficients in a ring. Euclidean domains, principal ideal domains and unique factorization domains. Factorization of polynomials with coefficients in a unique factorization domain. Irreducibility criteria for polynomials. Algebraically closed fields; the "fundamental theorem of algebra".
Course Language
Italian
More information
Students in the categories identified by the project on innovative teaching will have the opportunity to have receptions also electronically and by appointment at times to be agreed with the teacher, or to view the recordings of lessons from previous years.
