20410386 - AL110-ALGEBRA 1

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

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

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

Core Documentation

Script by the lecturer.

G.M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Decibel-Zanichelli, (1996)

I. N. Herstein, Algebra, Editori Riuniti, (2003)

Type of delivery of the course

Lectures in class on blackboard and exercise classes. In the case of an extension of the health emergency from COVID-19, all the provisions that regulate the methods of carrying out the teaching activities and student assessment will be implemented. In particular, the following methods will apply: streaming lectures on the platform Microsoft Teams.

Type 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.

teacher profile | teaching materials
teacher profile | teaching materials

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

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

Core Documentation

Script by the lecturer.

G.M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Decibel-Zanichelli, (1996)

I. N. Herstein, Algebra, Editori Riuniti, (2003)

Type of delivery of the course

Lectures in class on blackboard and exercise classes. In the case of an extension of the health emergency from COVID-19, all the provisions that regulate the methods of carrying out the teaching activities and student assessment will be implemented. In particular, the following methods will apply: streaming lectures on the platform Microsoft Teams.

Type 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.

teacher profile | teaching materials