THREE-DIMENSIONAL MODELING

Course objectives

General objectives: acquire the basic knowledge of Algebra related to topics of elementary number theory and group theory. Specific objectives: Knowledge and understanding: at the end of the course the student will have acquired the basic notions and results related to: 1) Modular arithmetic. 2) Group theory. Applying knowledge and understanding: at the end of the module the student will be able to autonomously handle the initial techniques of group theory and to solve simple problems of modular arithmetic Critical and judgmental skills: the student will have the basis for analyze the similarities and relationships with notions acquired in the courses of the first year with particular reference to topics concerning linear algebra and transformation groups. Communication skills: The learner will have the ability to communicate rigorously the ideas and contents presented in the course. Learning skills: the knowledge gained will allow one to study, individually or in subsequent courses, more advanced topics related to the main algebraic structures.

Channel 1
DANIELE VALERI Lecturers' profile

Program - Frequency - Exams

Course program
First part. Arithmetic on Z and modular Euclidean division, MCD, Euclidean algorithm and Bezout identity, prime numbers, fundamental theorem of arithmetic. Congruences, invertible elements of Z / mZ, Euler's function, Euler-Fermat's theorem, Fermat's Small Theorem, equations and systems of congruential equations, Chinese remainder theorem, RSA. Second part. Elements of group theory Groups, subgroups and normal subgroups, quotients, homomorphisms, homomorphism and isomorphism theorems, conjugated elements, Lagrange and Cayley theorems. Cyclic groups and their subgroups, dihedral groups, symmetric groups (writing of a permutation in disjoint cycles, even and odd class, conjugated permutations in the symmetric group). Direct and semi-direct product of groups. Finite p-groups. Finitely generated Abelian groups and their classification. Group actions on a set. Sylow theorems and applications.
Prerequisites
It useful for the student to know and master the basics imparted in the first courses of mathematical analysis and linear algebra.
Books
Israel Herstein, Algebra, Editori Riuniti Michael Artin, Algebra, Bollati Boringhieri S. Weintraub, Galois Theory Lecture Notes on Groups Theory by J. Milne
Frequency
Attending is not mandatory.
Exam mode
The exam aims to evaluate learning through a written test (consisting in solving problems of the same type as those carried out in the exercises) and an oral test (consisting in the discussion of the most relevant topics illustrated in the course). The written test will last about three hours and can be replaced by two intermediate tests, both lasting at least two hours, the first of which will take place in the middle and the second at the end of the course. The first intermediate test will focus mainly on the topics of arithmetic and group theory, the second on the remaining topics of the course. Students who have obtained a grade of not less than 18/30 in the written test (or the average of the two intermediate tests) are admitted to the oral exam. The minimum grade to pass the exam is 18/30. The student must demonstrate that he has acquired sufficient knowledge of the topics of all parts of the program. To achieve a score of 30/30 cum laude, the student must demonstrate that he has acquired excellent knowledge of all the topics covered during the course and be able to link them in a logical and coherent manner.
Lesson mode
The course consists of lectures and exercise sessions.
Channel 2
FRANCESCO MEAZZINI Lecturers' profile
  • Academic year2025/2026
  • CourseMathematics
  • CurriculumMatematica per le applicazioni
  • Year2nd year
  • Semester1st semester
  • SSDMAT/02
  • CFU6