Algebra DSBA 2020/2021

Schedule

• Lecture Thursday 16:20–17:40

• Seminar 201 Thursday 18:10–19:30

Teachers and assistants

Группа	201	202	203	204
Lecturer	Dima Trushin Telegram
Teacher	Dima Trushin	Andrew Mazhuga	Nikita Medved	Galina Kaleeva
Assistant	Masha Marchenko	Daniil Kopytov	Ваня Пешехонов	Dasha Ivanova

Consultations schedule

	Teacher/Assistant	Monday	Tuesday	Wednesday	Thursday	Friday
1	Dima Trushin			zoom since 17:00
2	Andrew Mazhuga
3	Nikita Medved			write me in telegram https://t.me/medvednikita and we will schedule a meeting
4	Galina Kaleeva			16.20-17.40. Offline: S 913 Online: Zoom Passcode: algebra In case I'm offline text me via telegram
5	Masha Marchenko
6	Daniil Kopytov
7	Vanya Peshekhonov				zoom c 18:20
8	Dasha Ivanova

Grading system

The final grade is computed as follows

F = 0,3 * H + 0,3 T + 0,4 E

where H is the grade for the home assignments, T is the written test grade, and E is the final exam grade.

Only the final grade is rounded in the final formula according to the standard rule.

Lecture abstracts

Lecture 1 (08.04.2021). Binary operations. Associativity, neutral element, inverse element, commutativity. Definition of a group. Additive and multiplicative notations. Subgroups and cyclic subgroups. The order of an element of a group.

Lecture 2 (15.04.2021). Subgroups of the group of integers. Left and right cosets, examples. Normal subgroups. The Lagrange theorem.

Lecture 3 (22.04.2021). Five corollaries of the Lagrange theorem. Homomorphisms and Isomorphisms of groups. Image and kernel of a homomorphism. Normal subgroups. Direct product of groups. Finite Abelian Groups. The Chinese Remainder Theorem. Structure of a finite abelian group.

Lecture 4 (29.04.2021). Second version of the Chinese Remainder Theorem. Structure of Z_{p^n}^*. Cryptography. Exponentiation by squaring (fast raising to a power algorithm). The discrete logarithm problem. Diffie-Hellman key exchange.

Lecture 5 (13.05.2021). Rings, commutative rings, fields, subrings. Invertible elements, zero divisors, nilpotent and idempotent elements. Ideals. Description of ideals in Z and Z_n. Homomorphisms and isomorphisms of rings. The Chinese remainder theorem for rings. The kernel and the image of a homomorphism, their properties.

Lecture 6 (20.05.2021). Polynomials in one variable. Euclidean algorithm, greatest common divisor, ideals of F[x]. Irreducible polynomials and unique factorization of polynomials in F[x]. Ring of remainders, the Chinese Remainder Theorem for polynomials.

Lecture 7 (27.05.2021). Characteristic of a field. Field extensions, an extension by a root. Finite fields: number of elements in a finite field, multiplicative group of a finite field is cyclic, classification of finite fields (without proof). How to produce finite fields. Galois random generator. Stream cipher.

Lecture 8 (03.06.2021). Polynomials in several variables. Lexicographical orders, stabilization of strictly descending chains of monomials. An elementary reduction, a reduction with respect to a set of polynomials, remainders, Groebner basis. Stabilization of reduction.

Lecture 9 (10.06.2021). S-polynomials and the Buchberger criterion. Ideals in a polynomial ring, the Buchberger algorithm to produce a Groebner basis of an ideal. A ring of remainders. Membership problem and variable elimination.

Lecture 10 (17.06.2021). The Diamond Lemma. A proof of the Buchberger criterion.

Problem sheets

The solutions should be sent to your teaching assistant before the beginning of the next seminar.

Seminar 1 (08.04.2021). Problems

Seminar 2 (15.04.2021). Problems

Seminar 3 (22.04.2021). Problems

Seminar 4 (29.04.2021). Problems

Seminar 5 (13.05.2021). Problems

Seminar 6 (20.05.2021). Problems

Since Homework 6 is late, there is a special deadline for Group 201. The new deadline is Sunday 23:00 May 30. Deadlines for the other groups remain the same.

Seminar 7 (27.05.2021). Problems

Seminar 8 (03.06.2021). Problems Deadline for group 204: Friday, 18.06.2021, 14.40.

Seminar 9 (10.06.2021). Problems

Test

Demo variant of the test

Exam

• The link to the rules and information regarding the exam.

Results

• Homework

201	202	203	204

• Summary Statement

201	202	203	204

Links

• Lecture Notes

• Seminars 201

• Two seminars 203 online: 8, 9 and corresponding whiteboards: 8, 9