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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