Unit ALGEBRA I
- Course
- Mathematics
- Study-unit Code
- GP006034
- Curriculum
- In all curricula
- Teacher
- Massimo Giulietti
- Teachers
-
- Massimo Giulietti
- Marco Timpanella (Codocenza)
- Hours
- 40 ore - Massimo Giulietti
- 7 ore (Codocenza) - Marco Timpanella
- CFU
- 6
- Course Regulation
- Coorte 2024
- Offered
- 2024/25
- 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
- The main goal of the teaching is to provide students with basic knowledge in the field of set theory and some algebraic structures, in order to then be able to undertake subsequent studies. Particular attention is given to the understanding of the arguments and to the rigor in the presentation of concepts and argumentations. Knowledge and comprehension: Mathematical comprehension of the proposed topics, knowledge of the theory developed on sets, functions, cardinality, congruences and fundamental examples discussed on these topics. Methods for verifying knowledge: Written and oral exam. Capacity: Being able to independently read and understand basic Algebra texts, connect arguments, find examples and counterexamples. Being able to produce simple rigorous proofs of mathematical results and problem solving of simple problems related to what has been illustrated in class. Methods for verifying skills: Written and oral exam. Autonomy of judgment: The display of the contents and arguments will be carried out in order to improve the ability of the student to recognize rigorous proofs Communication skills: The presentation of the topics will be organized to allow the acquisition of a good ability to communicate problems, ideas and solutions concerning Algebra, both in written and oral form.
- 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 and an oral test. The written test verifies the ability to produce rigorous demonstrations of problems and statements related to the topics of the course. The oral exam verifies the ability to clearly and rigorously explain some of the course contents. All written tests last about two hours and consist in solving some problems that can also be small parts of theory and serve to check the level of understanding of the topics covered and the ability to connect them. The written test of each session contains 4 problems, one on equivalence relations, one on posets, one on congruences, and one on complex numbers and/or induction principle. The oral exam, lasting about 30 minutes, tends to confirm the level of understanding of the topics covered and of critical study and personal re-elaboration. For information on support services for students with disabilities and / or SLD, visit the page http://www.unipg.it/disabilita-e-dsa
- 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.