Matematica | ALGEBRA I

cod. 0512300002

ALGEBRA I

	0512300002
	DIPARTIMENTO DI MATEMATICA

	MATHEMATICS
	2015/2016

CURRICULUM MATEMATICA AD INDIRIZZO APPLICATIVO

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2010
	SECONDO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/02	9	72	LESSONS	BASIC COMPULSORY SUBJECTS

	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.

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.