Università degli Studi di Pavia

Dipartimento di Matematica ''F. Casorati''

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Algebra 1

Professors:
Canonaco Alberto, Pirola Gian Pietro
Year:
2015/2016
Course code:
500202
ECTS:
9
SSD:
MAT/02
DM:
270/04
Lessons:
84
Period:
I semester
Language:
Italian

Objectives

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

Teaching methods

Lectures and exercise sessions

Examination

Written and oral exam

Prerequisites

The contents of the course of Linear Algebra.

Syllabus

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".

Bibliography

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

Modules

Professor:
Canonaco Alberto
Lessons:
56
ECTS:
6
SSD:
MAT/02

Professor:
Pirola Gian Pietro
Lessons:
28
ECTS:
3
SSD:
MAT/02


Dipartimento di Matematica ''F. Casorati''

Università degli Studi di Pavia - Via Ferrata, 5 - 27100 Pavia
Tel +39.0382.985600 - Fax +39.0382.985602