Unit ALGEBRA I
- Course
- Mathematics
- Study-unit Code
- GP006034
- Curriculum
- In all curricula
- Teacher
- Massimo Giulietti
- Teachers
-
- Massimo Giulietti
- Hours
- 47 ore - Massimo Giulietti
- CFU
- 6
- Course Regulation
- Coorte 2020
- Offered
- 2020/21
- Learning activities
- Caratterizzante
- Area
- Formazione teorica
- Academic discipline
- MAT/02
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Classical numerical sets: integers; rationals; reals; complex numbers. Prime numbers. Proofs by induction. Proofs ab absurdo.
Finite and infinite sets: properties and operations. Relations. Applications. Permutations. Cardinality of a set. Countable sets.
Combinatorial calculus. The ring of residue classes modulo an integer n. The Chinese Remainder Theorem. Basics on groups. - Reference texts
- Dikran N. Dikranjan e Maria Silvia Lucido, Aritmetica e algebra. Liguori Editore.
- Educational objectives
- Basis of modern mathematical language: sets, applications, relationships, systems, cardinals, etc. Methods in which a mathematical theory is developed starting from axioms, through the construction of the main numerical sets. First examples of finite algebraic structures obtained as quotients.
- Prerequisites
- Elementary algebra of the first two years in high school.
- Teaching methods
- Frontal lectures. All theoretical results will be rigorously proved and many related exercises will be proposed.
- Other information
- Learning verification modality
- The exam consists of a written test (or two progress assessments) and a final oral exam.
The written exam requires the solution of three/four problems (eigenvectors and diagonalization of matrices, reduction to canonical form of quadratic forms,
Euclidean affine spaces, topology) and it has a duration of 3 hours. Its objectives are to evaluate the resolutive capacity of the problems and the proper use of acquired knowledge.
The oral exam consists of a talk of about 30 minutes.
It is aimed at testing the degree of comprehension the students have reached, expositive skills and capacity of finding connections between the topics studied. If it is required, the exam can be taken in English. - Extended program
- Classical numerical sets: N, Z, Q and R.
Proofs ab absurdo and proofs by induction. The square root of a prime number is not rational.
The set C of complex numbers. Sum and product of complex numbers. Conjugate complex numbers. Reciprocate of a complex number.
Cartesian and trigonometric representation of a complex number.
Modulus and argument of a complex number.
De Moivre formula. n-th roots of unity in the complex field.
Fundamental Theorem of Algebra (without proof). Every algebraic equation of odd degree with real coefficients admits at least one real solution.
Elementary operations between sets. Cartesian product. The power-set of a set.
The power-set of a set of size n has size 2n.
Binomail coefficients. Tartaglia-Pascal triangle.
Applications. Injective, Surjective and Bijective applications.
Relations. Order relations and equivalence relations. Quotient set.
Countable sets. Cantor theorem about the countability of Q.
The power-set of a set X has cardinality strictly greater than the cardinality of X.
R is not countable.
Prime integers. Euclidean division. Euclid algorithm for determining the greatest common divisor between two integers. Bezout identity.
Euclid lemma: if a prime p divides the product of two integers, then p divides at least one of them.
The Fundamental Theorem of Arithmetic. Euclid theorem on the existence of infinitely many primes.
Congruences. Elementary properties. Congruence equations of the first degree. Diophantine equations.
Chinese Remainder Theorem. Criterions for divisibility by 3, 4, 9, 11. Little Fermat Theorem. Euler Phi function. Calculation of phi(n).
Euler Theorem. Wilson Theorem. The congruence x2=-1 (mod p) with p an odd prime has a solution if and only if p=1 (mod 4). The Diophantine equation x^2+y^2=n.
Pithagorean triples.
Algebraic structures. Semigroups, Monoids, Groups.
Some examples of abelian and non-abelian groups. The group of nxn invertible matrices.
The symmetric group S_n.
The Boolean group of the power-set of a set X.
Subgroups. Criterion for establishing whether a subset S of a group G is a subgroup of G.
Order o(x) of an element x of a group G.
The subgroup generated by x. If o(x)=n, then xh has order n/MCD(n,h).
For every element x of a multiplicative group G of order n, we have x^n=1.
Right and left cosets of a subgroup. Lagrange Theorem.
Definitions of ring and field.
Examples of rings and fields.