Matematica | ALGEBRA I / ALGEBRA II

Matematica ALGEBRA I / ALGEBRA II

 0512300038 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2021/2022

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2018 FULL ACADEMIC YEAR
 SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY 1 ALGEBRA I MAT/02 6 48 LESSONS BASIC COMPULSORY SUBJECTS 2 ALGEBRA II MAT/02 6 48 LESSONS BASIC COMPULSORY SUBJECTS

 MERCEDE MAJ21
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.
1. KNOWLEDGE AND UNDERSTANDING: THIS COURSE WILL PROVIDE AN INTRODUCTION TO THE SET THEORY, AND TO THE GENERAL THEORY OF ALGEBRAIC STRUCTURES, WITH EMPHASYS ON PROPERTIES OF GROUPS, RINGS AND VECTOR SPACES.
2. APPLYING KNOWLEDGE AND UNDERSTANDING: COURSE AIM IS ALSO TO ENABLE STUDENTS TO IDENTIFY FINITE AND INFINITE SETS OF ANY CARDINALITY, AND TO USE ALGEBRAIC STRUCTURES LIKE GROUPS, RINGS AND VECTOR SPACES. THE COURSE WILL TEND TO ENCOURAGE THE STUDENT'S ABILITY TO EXPOSE THE ACQUIRED KNOWLEDGE CLEARLY AND RIGOROUSLY.
ANOTHER GOAL OF THIS COURSE IS HABITING THE STUDENT TO FORMULATE PROBLEMS AND TO REASON STRICTLY.
THE STUDENT MUST BE ABLE TO ENCOUNTER DEFINITIONS, PROBLEMS AND THEOREMS REGARDING THE CONTENT OF THE TEACHING IN A CORRECT AND STRICT WAY. IT MUST ALSO BE ABLE TO SOLVE EXERCISES.
Prerequisites
BASIC KNOWLEDGE ACQUIRED THROUGH HIGH SCHOOL COURSES.
Contents
MODULE I

SET THEORY (16 HOURS). BASIC CONCEPTS. OPERATIONS ON SETS AND THEIR PROPERTIES. SET OF SUBSETS OF A SET. CARTESIAN PRODUCT AND CORRESPONDENCES BETWEEN SETS. FUNCTIONS. IMAGE AND COUNTERIMAGE AND THEIR PROPERTIES. INJECTIVE, SURJECTIVE, BIJECTIVE FUNCTIONS. FUNCTION COMPOSITION. INVERSE OF A BIECTIVE FUNCTION. RIGHT AND LEFT INVERSE. THE TWO FORMS OF THE PRINCIPLE OF MATHEMATICAL INDUCTION. DEFINITIONS BY INDUCTION. THE AXIOM OF CHOICE.

ELEMENTS OF COMBINATORIAL CALCULUS (6 HOURS). THE PRINCIPLE OF INCLUSION / EXCLUSION. NUMBER OF FUNCTIONS AND OF INJECTIVE FUNCTIONS BETWEEN FINITE SETS. NUMBER OF PERMUTATIONS OF A FINITE SET. NUMBER OF SUBSETS OF A FINITE SET. BINOMIAL COEFFICIENTS. THE BINOMIAL THEOREM.

EQUIVALENCE RELATIONS (4 HOURS). PARTITIONS. EQUIVALENCE CLASSES. FACTOR SET. THE FUNDAMENTAL THEOREM ON EQUIVALENCE RELATIONS.

CONGRUENCES BETWEEN INTEGERS (8 HOURS). DEFINITION, COMPATIBILITY WITH THE OPERATIONS. OPERATIONS WITH INTEGERS MODULO N. INVERSES MODULO N. LINEAR CONGRUENTIAL EQUATIONS.
THE CHINESE REMAINDER THEOREM.

ORDERING RELATIONS (6 HOURS). MINIMAL AND MAXIMAL ELEMENTS. MINIMUM AND MAXIMUM. MINORANTS AND MAJORANTS. GREATEST COMMON DIVISOR AND MINIMUM COMMON MULTIPLE. HASSE DIAGRAMS. TOTALLY ORDERED SETS. WELL ORDERED SETS. PATTERNS. INDUCTIVE SETS. ZORN’S LEMMA.

CARDINALITY THEORY (8 HOURS). EQUIPOTENT SETS. CARDINALITY OF A SET. COMPARISON BETWEEN CARDINALS. THEOREMS OF CANTOR, SCHRODER, BERNSTEIN, HARTOGS THEOREM (WITHOUT PROOF). TRICHOTOMY THEOREM. CANTOR THEOREM. CARDINALITY OF P(S). FINITE AND INFINITE SETS. CHARACTERIZATION THEOREM OF INFINITE SETS. COUNTABLE SETS. JOIN OF A SUCCESSION OF COUNTABLE SETS. CARDINALITY OF Z, Q AND R. THE POWER OF CONTINUUM. CONTINUUM HYPOTHESIS. GENERALIZED CONTINUUM HYPOTHESIS.

MODULE II

ALGEBRAIC STRUCTURES (8 HOURS). INTERNAL COMPOSITION LAWS, COMMUTATIVE LAWS, ASSOCIATIVE LAWS, STABLE SUBSETS, INDUCED OPERATIONS, NEUTRAL ELEMENT, INVERTIBLE ELEMENTS. STABLE SUBSET GENERATED BY A SUBSET. ASSOCIATIVITY THEOREM (WITHOUT PROOF). REGULAR ELEMENTS, FUNDAMENTAL THEOREM AND COROLLARY RELATED TO FINITE SETS. CONGRUENCES, QUOTIENT OPERATION. THE CONCEPT OF AN ALGEBRAIC STRUCTURE. HOMOMORPHISM THEOREM. EXTERNAL OPERATIONS.

ELEMENTS OF GROUP THEORY (20 HOURS). DEFINITION OF GROUP. SUBGROUPS AND THEIR CHARACTERIZATIONS. INTERSECTION OF SUBGROUPS, SUBGROUP GENERATED BY A SUBSET, SUBGROUP GENERATED BY THE UNION OF A FAMILY OF SUBGROUPS. GROUPS OF ORDER 6, SYMMETRIC GROUPS, GENERAL LINEAR GROUP OF DEGREE N ON A UNIT COMMUTATIVE RING. EQUIVALENCES IN A GROUP, INDEX OF A SUBGROUP, LAGRANGE THEOREM. NORMAL SUBGROUPS, FACTOR GROUP. SUBGROUPS OF A FACTOR GROUP. SUBGROUPS AND FACTORS OF Z. HOMOMORPHISM THEOREMS. CAYLEY'S THEOREM. CYCLIC GROUPS. PERIOD OF AN ELEMENT.

ELEMENTS OF RING THEORY (12 HOURS). DEFINITION, CALCULATION RULES, ZERO DIVISORS, INVERTIBLE ELEMENTS. INTEGRAL DOMAINS, DIVISION RINGS, FIELDS. SUBRINGS, DIVISION SUBRINGS. EXAMPLES. INTERSECTION OF SUBRINGS, SUBRING GENERATED BY A SUBSET. IDEALS OF A RING, UNIT RINGS WITHOUT NON-TRIVIAL IDEALS. MAXIMAL IDEALS. KRULL THEOREM. CONGRUENCES IN A RING, FACTOR RING, SUBRINGS AND IDEALS OF A FACTOR RING, FACTOR RING OVER A MAXIMAL IDEAL. HOMOMORPHISM THEOREMS. THE RING OF THE INTEGERS AND ITS FACTOR RINGS. CHARACTERISTIC OF A UNIT RING.

ELEMENTS OF THEORY OF LEFT AND RIGHT VECTOR SPACES ON A DIVISION RING (8 HOURS). VECTOR SPACES ON FIELDS. CALCULATION RULES. SUBSPACES. SUBSPACE GENERATED BY A SUBSET. LINEAR COMBINATIONS. LINEAR DEPENDENCE. FREE SUBSETS. BASES. BASIC EXISTENCE THEOREM. DIMENSION. FACTOR SPACES. HOMOMORPHISMS OF VECTOR SPACES. HOMOMORPHISM THEOREMS.
Teaching Methods
ALGEBRA I/II COURSE INCLUDES 96 HOURS OF CLASSROOM TEACHING, DIVIDED IN TWO MODULES.
COURSE ATTENDANCE IS STRONGLY RECOMMENDED EVEN IF NOT MANDATORY.
LECTURES WILL INCLUDE THEORETICAL TOPICS SUPPORTED BY PRESENTATION OF EXAMPLES AND EXERCISES TO CLARIFY METHODS AND CONTEXTS OF USE OF THE ADDRESSED TOPICS. FOR THIS REASON, EXERCISES ARE INTEGRATED INTO THE SCHEDULED LESSONS.
SOME LESSONS EXCLUSIVELY DEDICATED TO EXERCISES' DISCUSSION ARE ALSO PLANNED AT THE END OF EACH MODULE AND WITHIN THE 96 HOURS OF TEACHING.

Verification of learning
THE AIM OF THE EXAMINATION IS TO EVALUATE THE FAMILIARITY OF THE STUDENT WITH THE MATHEMATICAL LANGUAGE, THE ABSTRACT FORMULATION OF A PROBLEM AND WITH THE STRICT REASONING. FOR BOTH MODULES, STUDENT HAVE TO PASS A WRITTEN EXAM AND AN ORAL INTERVIEW.

THE FIRST WRITTEN EXAM, WHICH CAN ALSO BE TAKEN AT THE END OF THE FIRST MODULE, INCLUDES EXERCISES ON SET THEORY, COMBINATORIAL CALCULUS, CONGRUENCES BETWEEN INTEGERS AND ORDER RELATIONS. IN CASE OF SATISFACTORY RESULTS OF THE TEST (WITH MINIMUM VOTE OF 18/30) THE STUDENT WILL TAKE THE FIRST ORAL INTERVIEW, BY WHICH THE KNOWLEDGE ACQUIRED ON THE TOPICS COVERED IN THE FIRST MODULE WILL BE ASSESSED.

THE SECOND WRITTEN EXAM INCLUDES EXERCISES ON GROUPS, RINGS AND VECTOR SPACES. IN CASE OF SATISFACTORY RESULTS OF THE TEST (WITH MINIMUM VOTE OF 18/30), THE STUDENT WILL TAKE A SECOND ORAL INTERVIEW BY WHICH THE KNOWLEDGE ACQUIRED ABOUT THE TOPICS COVERED IN THE SECOND MODULE WILL BE ASSESSED.

IN THE FINAL EVALUATION (OUT OF 30), THE WRITTEN AND ORAL EXAMS RELATED TO THE TWO MODULES WILL COUNT IN AN EQUAL WAY. THE EXCELLENCE MAY BE REACHED BY STUDENTS WHO PROVE THEIR ABILITY TO APPLY THE ACQUIRED KNOWLEDGE AND SKILLS EVEN IN DIFFERENT CONTEXTS WITH RESPECT TO THOSE PROPOSED DURING THE LESSONS.
Texts
M. CURZIO, P. LONGOBARDI, M. MAJ "LEZIONI DI ALGEBRA", LIGUORI, NAPOLI, II EDIZIONE 2014
M.CURZIO, P. LONGOBARDI, M. MAJ. "ESERCIZI DI ALGEBRA - UNA RACCOLTA DI PROVE D'ESAME SVOLTE", LIGUORI, NAPOLI, II EDIZIONE 2011.