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
teacher profile teaching materials
-Sets and elements
-Propositional logic
-Subsets, union, intersection and complement
-Power set and partitions
-Cartesian product
Correspondences and relations
-Correspondences
- Order relations
- Equivalence relations
Functions
- Generalities on functions
- Composite functions
- Inverse functions
- Kernel relation and decomposition theorem
Natural numbers and cardinality
- The set of natural numbers and induction
- The cardinality of a set
The ring of integers
-Construction of the set of whole numbers
- Generalities about rings
- The Euclidean division
- The fundamental theorem of arithmetic
The rings of residue classes
- Definition and first properties
- Linear congruences and systems of linear congruences
-Morphisms
-The Fermat's little Theorem and Euler's theorem
The field of rational numbers
-Construction of the set of rational numbers
- The positional notation of rational numbers
Polynomials
- Generalities on polynomials
- Roots, division and factorization of polynomials
- Polynomials with integer and rational coefficients
The fields of real numbers and complex numbers
- Notions on the construction of the reals
-Positional writing of real numbers
- Definition of the complex field
-Polinomials with real and complex coefficients
- Algebraic numbers and transcendental numbers
- Polar or trigonometric form of complex numbers
- Roots of unity and cyclotomic polynomials
More information on: https://sites.google.com/site/al11020192020/
G.M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Decibel-Zanichelli, (1996)
M. Fontana - S. Gabelli: Insiemi, numeri e polinomi. Primo ciclo di lezioni del corso di Algebra con esercizi svolti. CISU, (1989)
More information on: https://sites.google.com/site/al11020192020/
Programme
The language of sets-Sets and elements
-Propositional logic
-Subsets, union, intersection and complement
-Power set and partitions
-Cartesian product
Correspondences and relations
-Correspondences
- Order relations
- Equivalence relations
Functions
- Generalities on functions
- Composite functions
- Inverse functions
- Kernel relation and decomposition theorem
Natural numbers and cardinality
- The set of natural numbers and induction
- The cardinality of a set
The ring of integers
-Construction of the set of whole numbers
- Generalities about rings
- The Euclidean division
- The fundamental theorem of arithmetic
The rings of residue classes
- Definition and first properties
- Linear congruences and systems of linear congruences
-Morphisms
-The Fermat's little Theorem and Euler's theorem
The field of rational numbers
-Construction of the set of rational numbers
- The positional notation of rational numbers
Polynomials
- Generalities on polynomials
- Roots, division and factorization of polynomials
- Polynomials with integer and rational coefficients
The fields of real numbers and complex numbers
- Notions on the construction of the reals
-Positional writing of real numbers
- Definition of the complex field
-Polinomials with real and complex coefficients
- Algebraic numbers and transcendental numbers
- Polar or trigonometric form of complex numbers
- Roots of unity and cyclotomic polynomials
More information on: https://sites.google.com/site/al11020192020/
Core Documentation
Script by the lecturer.G.M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Decibel-Zanichelli, (1996)
M. Fontana - S. Gabelli: Insiemi, numeri e polinomi. Primo ciclo di lezioni del corso di Algebra con esercizi svolti. CISU, (1989)
More information on: https://sites.google.com/site/al11020192020/
Reference Bibliography
Script by the lecturer. G.M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Decibel-Zanichelli, (1996) M. Fontana - S. Gabelli: Insiemi, numeri e polinomi. Primo ciclo di lezioni del corso di Algebra con esercizi svolti. CISU, (1989) More information on: https://sites.google.com/site/al11020192020/Type of delivery of the course
Lectures in class on blackboard and exercise classesType of evaluation
Written and oral exams. Two tests during the semester can replace the written exam. All written tests contain six exercises to be solved in 3 hours. More information on: https://sites.google.com/site/al11020192020/ teacher profile teaching materials
-Sets and elements
-Propositional logic
-Subsets, union, intersection and complement
-Power set and partitions
-Cartesian product
Correspondences and relations
-Correspondences
- Order relations
- Equivalence relations
Functions
- Generalities on functions
- Composite functions
- Inverse functions
- Kernel relation and decomposition theorem
Natural numbers and cardinality
- The set of natural numbers and induction
- The cardinality of a set
The ring of integers
-Construction of the set of whole numbers
- Generalities about rings
- The Euclidean division
- The fundamental theorem of arithmetic
The rings of residue classes
- Definition and first properties
- Linear congruences and systems of linear congruences
-Morphisms
-The Fermat's little Theorem and Euler's theorem
The field of rational numbers
-Construction of the set of rational numbers
- The positional notation of rational numbers
Polynomials
- Generalities on polynomials
- Roots, division and factorization of polynomials
- Polynomials with integer and rational coefficients
The fields of real numbers and complex numbers
- Notions on the construction of the reals
-Positional writing of real numbers
- Definition of the complex field
-Polinomials with real and complex coefficients
- Algebraic numbers and transcendental numbers
- Polar or trigonometric form of complex numbers
- Roots of unity and cyclotomic polynomials
More information on: https://sites.google.com/site/al11020192020/
G.M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Decibel-Zanichelli, (1996)
M. Fontana - S. Gabelli: Insiemi, numeri e polinomi. Primo ciclo di lezioni del corso di Algebra con esercizi svolti. CISU, (1989)
More information on: https://sites.google.com/site/al11020192020/
Programme
The language of sets-Sets and elements
-Propositional logic
-Subsets, union, intersection and complement
-Power set and partitions
-Cartesian product
Correspondences and relations
-Correspondences
- Order relations
- Equivalence relations
Functions
- Generalities on functions
- Composite functions
- Inverse functions
- Kernel relation and decomposition theorem
Natural numbers and cardinality
- The set of natural numbers and induction
- The cardinality of a set
The ring of integers
-Construction of the set of whole numbers
- Generalities about rings
- The Euclidean division
- The fundamental theorem of arithmetic
The rings of residue classes
- Definition and first properties
- Linear congruences and systems of linear congruences
-Morphisms
-The Fermat's little Theorem and Euler's theorem
The field of rational numbers
-Construction of the set of rational numbers
- The positional notation of rational numbers
Polynomials
- Generalities on polynomials
- Roots, division and factorization of polynomials
- Polynomials with integer and rational coefficients
The fields of real numbers and complex numbers
- Notions on the construction of the reals
-Positional writing of real numbers
- Definition of the complex field
-Polinomials with real and complex coefficients
- Algebraic numbers and transcendental numbers
- Polar or trigonometric form of complex numbers
- Roots of unity and cyclotomic polynomials
More information on: https://sites.google.com/site/al11020192020/
Core Documentation
Script by the lecturer.G.M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Decibel-Zanichelli, (1996)
M. Fontana - S. Gabelli: Insiemi, numeri e polinomi. Primo ciclo di lezioni del corso di Algebra con esercizi svolti. CISU, (1989)
More information on: https://sites.google.com/site/al11020192020/
Reference Bibliography
Script by the lecturer. G.M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Decibel-Zanichelli, (1996) M. Fontana - S. Gabelli: Insiemi, numeri e polinomi. Primo ciclo di lezioni del corso di Algebra con esercizi svolti. CISU, (1989) More information on: https://sites.google.com/site/al11020192020/Type of delivery of the course
Lectures in class on blackboard and exercise classesType of evaluation
Written and oral exams. Two tests during the semester can replace the written exam. All written tests contain six exercises to be solved in 3 hours. More information on: https://sites.google.com/site/al11020192020/