Бакалавриат
2026/2027
Алгебра
Статус:
Курс обязательный (Прикладной анализ данных)
Где читается:
Факультет компьютерных наук
Когда читается:
1-й курс, 4 модуль
Охват аудитории:
для своего кампуса
Язык:
английский
Кредиты:
3
Контактные часы:
40
Course Syllabus
Abstract
The course is a gentle introduction to the theory of algebraic structures, including Groups, Rings, Fields, and Modules, with some applications Algorithms. The course provides professional background for further courses related to Discrete Mathematics, Formal Languages, Game Theory and Information Security.
Prerequisites: the course is based on knowledge of numerical systems and functions studied in high school, as well as on the basic concepts of the courses on Linear Algebra and Geometry, Calculus and Discrete Mathematics in the first three modules.
Learning Objectives
- Introduction of main algebraic structures with explicit examples, motivations and applications.
- Practice in basic computations with groups, rings, fields and modules.
- Forming students’ skills in using matrix and polynomial techniques to formalize and solve applied problems, including economic and financial ones.
- Study of some applications of the theory of finite groups and finite fields.
Expected Learning Outcomes
- Apply the Homomorphism theorem to describe a quotient group.
- Check if a given group is cyclic or not.
- Check if a given subgroup is normal.
- Check if a set with an operation is a group or not.
- Compute explicitly a finite extension of a field.
- Compute orders of all elements of a given group.
- Construct a quotient group for a given group and a subgroup.
- Describe key exchange algorithm.
- Describe multiplication and addition of the ring of remainders.
- Describe the division algorithm in a polynomial ring with one variable.
- Find the characteristic of a given field.
- Formulate a criterion for the ring of remainders to be a field.
- Give an example of a field.
- Give an example of a finite field.
- Give an example of a group.
- Give an example of a normal subgroup.
- Give an example of a ring.
- Give an example of an ideal.
- Give an example of an irreducible and reducible polynomials.
- Give the definition of a characteristic of a field.
- Give the definition of a field.
- Give the definition of a finite field.
- Find the center and the commutator subgroup of a group.
- Give an example of a module which is not a vector space.
- Describe the commutator subgroup of a given group.
Bibliography
Recommended Core Bibliography
- David A. Cox, John Little, Donal O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. - Springer International Publishing, Switzerland, 2015. Print ISBN: 978-3-319-16720-6. Online ISBN: 978-3-319-16721-3.
Recommended Additional Bibliography
- 9781466570276 - Katz, Jonathan; Lindell, Yehuda - Introduction to Modern Cryptography, 2nd Edition - 2014 - CRC Press - http://search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=1766746 - nlebk - 1766746