# Matematica | ALGEBRA I / ALGEBRA II

## Matematica ALGEBRA I / ALGEBRA II

 0512300038 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2022/2023

 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 4 32 LESSONS BASIC COMPULSORY SUBJECTS MAT/02 2 24 EXERCISES BASIC COMPULSORY SUBJECTS 2 ALGEBRA II MAT/02 4 32 LESSONS BASIC COMPULSORY SUBJECTS MAT/02 2 24 EXERCISES BASIC COMPULSORY SUBJECTS

 COSTANTINO DELIZIA21 T
ExamDate
ALGEBRA I / ALGEBRA II18/04/2023 - 13:00
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 (12 HOURS LESSON + 5 HOURS PRACTICE). BASIC CONCEPTS. OPERATIONS ON SETS AND THEIR PROPERTIES. SET OF SUBSETS. CARTESIAN PRODUCT AND RELATIONS. FUNCTIONS. IMAGE AND COUNTERIMAGE AND THEIR PROPERTIES. INJECTIVE, SURJECTIVE, BIJECTIVE FUNCTIONS. FUNCTION COMPOSITION. INVERSE FUNCTION. RIGHT AND LEFT INVERSE. THE TWO FORMS OF THE PRINCIPLE OF MATHEMATICAL INDUCTION. DEFINITIONS BY INDUCTION.
COMBINATORICS (4 HOURS LESSON + 2 HOURS PRACTICE). THE PRINCIPLE OF INCLUSION / EXCLUSION. NUMBER OF FUNCTIONS AND OF INJECTIVE FUNCTIONS BETWEEN FINITE SETS. NUMBER OF PERMUTATIONS OF A FINITE SET. BINOMIAL COEFFICIENTS. THE BINOMIAL THEOREM.
BINARY RELATIONS (13 HOURS LESSON + 6 HOURS PRACTICE). EQUIVALENCE RELATIONS. PARTITIONS. EQUIVALENCE CLASSES. FACTOR SET. THE FUNDAMENTAL THEOREM ON EQUIVALENCE RELATIONS.
CONGRUENCES BETWEEN INTEGERS. DEFINITION, ARITHMETICS MODULO N. LINEAR CONGRUENTIAL EQUATIONS. THE CHINESE REMAINDER THEOREM.
ORDERING RELATIONS. 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.
ALGEBRAIC STRUCTURES I (4 HOURS LESSON + 2 HOURS PRACTICE). 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.
CARDINALITY THEORY I (3 HOURS LESSON + 1 HOURS PRACTICE). EQUIPOTENT SETS. CARDINALITY OF A SET. COMPARISON BETWEEN CARDINALS. THEOREMS OF CANTOR-SCHRODER-BERNSTEIN AND OF HARTOGS THEOREM (WITHOUT PROOF). TRICHOTOMY THEOREM. CANTOR THEOREM. FINITE AND INFINITE SETS. COUNTABLE SETS. CARDINALITY OF THE SETS OF INTEGER NUMBERS AND OF RATIONAL NUMBERS. THE AXIOM OF CHOICE.
MODULE II
ALGEBRAIC STRUCTURES II (12 HOURS LESSON + 5 HOURS PRACTICE). CONGRUENCES, QUOTIENT OPERATION. THE CONCEPT OF AN ALGEBRAIC STRUCTURE. HOMOMORPHISM THEOREM. EXTERNAL OPERATIONS.
CARDINALITY THEORY II (12 HOURS LESSON + 5 HOURS PRACTICE). CHARACTERIZATION THEOREM OF INFINITE SETS. COUNTABLE SETS. CARDINALITY OF THE SETS OF REAL NUMBERS AND OF COMPLEX NUMBERS. THE POWER OF CONTINUUM. CONTINUUM HYPOTHESIS. GENERALIZED CONTINUUM HYPOTHESIS.
GROUP THEORY (12 HOURS LESSON + 5 HOURS PRACTICE). 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 GROUPS. 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.
RING THEORY (12 HOURS LESSON + 5 HOURS PRACTICE). 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, 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. CHARACTERISTIC OF A UNIT RING.
THEORY OF LEFT AND RIGHT VECTOR SPACES ON A DIVISION RING (12 HOURS LESSON + 5 HOURS PRACTICE). 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
THE COURSE INCLUDES 112 HOURS OF CLASSROOM TEACHING, DIVIDED IN TWO MODULES. FOR EACH MODULE THERE ARE 32 HOURS OF LESSONS AND 24 HOURS OF PRACTICE. COURSE ATTENDANCE IS STRONGLY RECOMMENDED EVEN IF NOT MANDATORY. THEORETICAL LESSONS WILL BE CONSTANTLY ALTERNATED WITH HOURS OF PRACTICE, DURING WHICH EXAMPLES AND EXERCISES WILL BE PRESENTED THAT ILLUSTRATE METHODS AND CONTEXTS OF USE OF WHAT EXPLAINED.

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 TEST AND AN ORAL INTERVIEW.

THE FIRST WRITTEN TEST, 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 TEST 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.