2021/2022

2022/2023

DM270

MAT/02 (ALGEBRA)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

2°

1st semester (29/09/2022 - 13/01/2023)

9

84 lesson hours

Italian

WRITTEN AND ORAL TEST

CANONACO ALBERTO (titolare) - 8 ECTS
FAVALE FILIPPO FRANCESCO - 1 ECTS

The contents of the course of Linear Algebra.

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.

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".

Lectures and exercise sessions

Notes provided by the teachers.
I.N. Herstein: "Algebra", Editori Riuniti.
M. Artin: "Algebra", Bollati Boringhieri.

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.