Università Roma Tre

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

Script by the lecturer.

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

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