Provide the elements of the "mathematical language" (set theory, elementary logic, numerical sets) and the knowledge of the basic tools of modern algebra (notions of operation, group, ring, field) through the development of examples that provide the motivations.
Curriculum
scheda docente
materiale didattico
I. Herstein, Algebra - Editori Riuniti (2010)
G.M. Piacentini Cattaneo, Algebra,un approccio algoritmico, Decibel -Zanichelli.
Programma
SETS AND FUNCTIONS. EQUIVALENCE RELATIONS. NATURAL NUMBERS. PEANO AXIOMS. THE PRINCIPLE OF INDUCTION. WELL ORDERING. CONSTRUCTIONS OF THE SET OF RELATIVE INTEGER NUMBERS AND OF THE SET OF RATIONAL NUMBERS. BASIC PROPERTIES OF COMPLEX NUMBERS. DIVISIBILITY IN THE INTEGERS, EUCLIDEAN ALGORITHM, GCD. DEFINITIONS AND EXAMPLES OF THE MAIN ALGEBRAIC STRUCTURES: GROUPS, RINGS, AND FIELDS. GROUP OF THE UNITS OF A RING. GROUPS OF PERMUTATIONS. THE RING OF INTEGERS MODULO N. LINEAR CONGRUENCES. EULER PHI FUNCTION. POLYNOMIAL RINGS WITH COEFFICIENTS IN RING OF NUMBERS: CONSTRUCTION, BASIC PROPERTIES, DIVISIBILITY, IRREDUCIBILITY CRITERIA, GAUSS LEMMA AND PRIMITIVE POLYNOMIALS.Testi Adottati
D. Dikranjan - M.S. Lucido, Aritmetica e algebra, Liguori.I. Herstein, Algebra - Editori Riuniti (2010)
G.M. Piacentini Cattaneo, Algebra,un approccio algoritmico, Decibel -Zanichelli.
Modalità Erogazione
Lectures by the teacher with sessions of exercises only. In any case, the instructions of the University regarding the possibility of transmitting the lessons on Microsoft Teams will be followed if this becomes necessary for the Covid emergency. It is required that students will discuss some chosen topics as a seminar.Modalità Frequenza
Attending is not mandatory but strongly recommended in personModalità Valutazione
The exam will consist of a written and an oral test at the end of the course. During the course there will be two partial written tests that will be evaluated as a written exam. There will be four tests (written and oral). The written test (including the partial tests) consists of 5/6 practical / theoretical exercises to be carried out in 2.30 / 3 hours.
scheda docente
materiale didattico
I. Herstein, Algebra - Editori Riuniti (2010)
G.M. Piacentini Cattaneo, Algebra,un approccio algoritmico, Decibel -Zanichelli.
Programma
SETS AND FUNCTIONS. EQUIVALENCE RELATIONS. NATURAL NUMBERS. PEANO AXIOMS. THE PRINCIPLE OF INDUCTION. WELL ORDERING. CONSTRUCTIONS OF THE SET OF RELATIVE INTEGER NUMBERS AND OF THE SET OF RATIONAL NUMBERS. BASIC PROPERTIES OF COMPLEX NUMBERS. DIVISIBILITY IN THE INTEGERS, EUCLIDEAN ALGORITHM, GCD. DEFINITIONS AND EXAMPLES OF THE MAIN ALGEBRAIC STRUCTURES: GROUPS, RINGS, AND FIELDS. GROUP OF THE UNITS OF A RING. GROUPS OF PERMUTATIONS. THE RING OF INTEGERS MODULO N. LINEAR CONGRUENCES. EULER PHI FUNCTION. POLYNOMIAL RINGS WITH COEFFICIENTS IN RING OF NUMBERS: CONSTRUCTION, BASIC PROPERTIES, DIVISIBILITY, IRREDUCIBILITY CRITERIA, GAUSS LEMMA AND PRIMITIVE POLYNOMIALS.Testi Adottati
D. Dikranjan - M.S. Lucido, Aritmetica e algebra, Liguori.I. Herstein, Algebra - Editori Riuniti (2010)
G.M. Piacentini Cattaneo, Algebra,un approccio algoritmico, Decibel -Zanichelli.
Modalità Erogazione
Lectures by the teacher with sessions of exercises only. In any case, the instructions of the University regarding the possibility of transmitting the lessons on Microsoft Teams will be followed if this becomes necessary for the Covid emergency. It is required that students will discuss some chosen topics as a seminar.Modalità Frequenza
Attending is not mandatory but strongly recommended in personModalità Valutazione
The exam will consist of a written and an oral test at the end of the course. During the course there will be two partial written tests that will be evaluated as a written exam. There will be four tests (written and oral). The written test (including the partial tests) consists of 5/6 practical / theoretical exercises to be carried out in 2.30 / 3 hours.