Matematica | ALGEBRA I

Matematica ALGEBRA I

 0512300002 DIPARTIMENTO DI MATEMATICA MATHEMATICS 2015/2016

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2010 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/02 9 72 LESSONS BASIC COMPULSORY SUBJECTS
 MERCEDE MAJ T
Objectives
THE AIM OF THIS COURSE IS TO MAKE STUDENTS FAMILIAR WITH THE MATHEMATICAL LANGUAGE, THE ABSTRACT FORMULATION OF A PROBLEM AND WITH THE STRICT REASONING.
WE WILL STUDY ALGEBRAIC STRUCTURES AND , IN PARTICULAR, GROUPS, RINGS AND VECTOR SPACES.
Prerequisites
BASIC KNOWLEDGE ACQUIRED THROUGH HIGH SCHOOL COURSES.
Contents
SET THEORY. SET OPERATIONS. RELATIONS. FUNCTIONS. EQUIVALENCE RELATIONS, PARTITIONS. PARTIAL ORDERS, TOTAL ORDERS, WELL-ORDERS. CHOICE AXIOM, ZORN LEMMA. NATURAL NUMBERS AND MATHEMATICAL INDUCTION. INTEGERS, CONGRUENCES.
ALGEBRAIC STRUCTURES: EXAMPLES, SUBSTRUCTURES, CONGRUENCES, HOMOMORPHISMS.
GROUPS: EXAMPLES, PERMUTATION GROUPS, MATRICES, SUBGROUPS, SUBGROUP GENERATED, LAGRANGE THEOREM, CONGRUENCES IN A GROUP, FACTOR GROUP, HOMOMORPHISMS, CAYLEY THEOREM, CYCLIC GROUPS, ORDER OF AN ELEMENT.
RINGS: EXAMPLES, POLYNOMIAL RINGS, SUBRINGS AND IDEALS, KRULL THEOREM, HOMOMORPHISMS, FACTOR RINGS, CHARACTERISTIC, RING OF QUOTIENTS.
VECTOR SPACES: EXAMPLES, SUBSPACES, HOMOMORPHISMS, BASIS, DIMENSION OF A VECTOR SPACE, DIRECT SUMMANDS, FINITE DIMENSIONAL VECTOR SPACES.
Teaching Methods
LECTURES.
EXERCISES.
ATTENDANCE TO CLASS LESSONS IS STRONGLY RECOMMENDED.
Verification of learning
WRITTEN AND ORAL EXAMINATION
Texts
M. CURZIO, P. LONGOBARDI, M. MAJ "LEZIONI DI ALGEBRA", LIGUORI, NAPOLI, 1994, I RISTAMPA 1996.

M.CURZIO, P. LONGOBARDI, M. MAJ. "ESERCIZI DI ALGEBRA - UNA RACCOLTA DI PROVE D'ESAME SVOLTE", LIGUORI, NAPOLI, II EDIZIONE 2011.