Topics in Discrete Mathematics
Code
8503
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
3.0
Teacher in charge
Jorge Manuel Leocádio André, Maria Helena Coutinho Gomes Almeida Santos
Weekly hours
3
Teaching language
Português
Subject matter
1. Divisibility in the ring of integers
Euclidian division, Euclides'''' algorythm, Bezout''''s Theorem and Euclides'''' Lemma, prime numbers and the Fundamental Theorem of Arythmetic.
2. Revisions in Algebraic structures
Groups, rings, normal subgroups and ideals, quotient groups and rings.
3. Congruences
Systems of residues, Euler''''s Theorem and Fermat''''s little Theorem, Wilson''''s Theorem, linear congruences and the Chinese remainder Theorem.
4. Arithmetic functions
Convolution product, Euler''''s function and other classical results.
5. Quadratic reciprocity
Legendre''''s symbol, Gauss'''' Lemma, Quadratic reciprocity law, quadratic congruences, second degree polynomials with coefficients in Zn.
6. Waring''''s problem
Bibliography
-Introdução à Teoria de Números, Filipe Oliveira, FCT - Universidade Nova de Lisboa
-Elementary Number Theory, Gareth Jones and Josephine Jones, Springer Undergraduate Mathematics Series
-An introduction to the Theory of Numbers, G.H. Hardy, Oxford Science Publication
-Introduction à la théorie des nombres, JM De Koninck & A Mercier, Collection Universitaire de Mathématiques, 1994.
Teaching method
During the Theoretical-Practical classes, the different contents of this course will be exposed. Students will be asked to solve exercises and elaborate proofs of some of the different resuts presented.
Any questions or doubts will be adressed during the classes, during the weeekly sessions specially programmed to it or even at special sessions previously arranged between professors and students.