Algebra DSBA 2018/2019

Материал из Wiki - Факультет компьютерных наук
Версия от 18:48, 16 июня 2019; Mednik (обсуждение | вклад)

(разн.) ← Предыдущая | Текущая версия (разн.) | Следующая → (разн.)
Перейти к: навигация, поиск

Teachers and assistants

Группа 181 182 183
Lecturer Ivan Arzhantsev
Teacher Ivan Arzhantsev Roman Avdeev Nikita Medved

mednik at

Assistant Danila Kutenin kutdanila at Maksim Siplivyj Ildus Sadrtdinov

Consultations schedule

Teacher/Assistant Monday Tuesday Wednesday Thursday Friday
Ivan Arzhantsev 17:00–18:30, room 603
Roman Avdeev 15:40–17:40, room 623 15:40–16:30, 18:10–19:00, room 623
Nikita Medved 16:40–18:00, room 623 18:10 (if you write me beforehand)
Danila Kutenin 12:00-13:00, room (each time telegram announcement)
Maksim Siplivyj 16:40–18:00
Ildus Sadrtdinov 16:40–18:00

Grading system

The cumulative grade is computed as follows:

C = 0,6 * H + 0,4 * T,

where H is the grade for the home assignments and T is the written test grade.

The final course grade is given by

F = 0,5 * C + 0,5 * E

where E is the final exam grade.

Grades in all formulas are rounded according to the standard rule.

Lecture abstracts

Lecture 1 (2.04.2019). Semigroups and groups: definitions and examples. Permutation groups and matrix groups. Subgroups. The order of an element and cyclic subgroups.

Lecture 2 (9.04.2019). Lagrange's theorem and its corollaries. Normal subgroups. Homomorphisms and isomorphisms. A classification of cyclic groups. Factor groups and the Homomorphism theorem.

Lecture 3 (16.04.2019). The homomorphism theorem. The center and direct products of groups. Theorem on factorization of direct products and factorization of finite cyclic groups.

Lecture 4 (23.04.2019). Free abelian groups and their subgroups. Stacked bases. An algorithm for transforming an integer matrix to a diagonal form. Classification of finite abelian groups. The exponent of a finite abelian group.

Lecture 5 (30.04.2019). Actions of a group on a set. Orbits and stabilizers. Transitive actions and free actions. Three actions of a group on itself. Conjugacy classes. Cayley's Theorem.

Lecture 6 (14.05.2019). Rings and fields. Zero divisors, invertible elements, nilpotents and idempotents. Ideals. Principal ideals. Factor rings and the Homomorphism Theorem.

Lecture 7 (21.05.2019). Polynomials in several variables. Symmetric polynomials. The lexicographic order. Elementary symmetric polynomials. The main theorem on symmetric polynomials. Vieta's formulas. The discriminant.

Lecture 8 (28.05.2019). Polynomials in one variable over a field. Greatest common divisor. Irreducible polynomials. Unique factorization property. Description of ideals. Properties of factor rings.

Lecture 9 (04.06.2019). The characteristic of a field. Extensions of fields. Finite extensions and their degrees. Algebraic and transcendental elements. The minimal polynomial of an algebraic element.

Lecture 10 (11.06.2019). Decomposition of a polynomial into linear factors. Finite fields. Cyclicity of the multiplicative group. Irreducible polynomials over the field $\ZZ_p$. The field with four elements.

Problem sheets

The Nth problem sheet contains the Nth homework.

Problems to lecture 1

Problems to lecture 2

Problems to lecture 3

Problems to lecture 4

Problems to lecture 5

Problems to lecture 6

Problems to lecture 7

Problems to lecture 8

Problems to lecture 9

Written test

The test has been on Tuesday 11.06.2019, 16:40-19:30. You could use any printed or handwritten notes, a non-programmable calculator.

Problems from the test


The exam will be oral.

List of topics


181 182 183

Reading list


  • Э.Б.Винберг. Курс алгебры. М.: МЦНМО, 2014 (English transl.: Ernest Vinberg. A Course in Algebra. Graduate Studies in Math. 56, Amer. Math. Soc., 2003)
  • Сборник задач по алгебре под редакцией А.И.Кострикина. Новое издание. М.: МЦНМО, 2015 (English transl.: Exercises in Algebra. Edited by A. Kostrikin, CRC Press, 1996)


Serge Lang. Algebra. Revised Third Edition. Graduate Texts in Math. 211, Springer, 2002