2018/2019

2019/2020

DM270

MAT/02 (ALGEBRA)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

2°

1st semester (30/09/2019 - 10/01/2020)

9

84 lesson hours

Italian

WRITTEN AND ORAL TEST

FREDIANI PAOLA (titolare) - 9 ECTS

The contents of the course of Linear Algebra.

The course is an introduction to some fundamental algebraic structures: groups, rings and fields.

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.