DSBA Algebra 2023 2024

Teachers and assistants

Группа	231	232	233	234	235
Lecturer	Dima Trushin Telegram
Teacher	Kirill Shakhmatov	Galina Kaleeva	Dima Trushin	Andrew Mazhuga	Dima Trushin
Assistant	Platon Chevychelov	Alexander Ukhanov	Roman Bokhyan	Kirill Belyakov	Rina Zyubanova

Consultations schedule

	Teacher/Assistant	How to contact	When
1	Dima Trushin	telegram	Thursday since 17:00 S812
2	Andrew Mazhuga	telegram
3	Kirill Shakhmatov
4	Galina Kaleeva	telegram, check the course chat	Office hours: Tuesday, 18.00, zoom (the link is in the course chat)
5	Alexander Ukhanov	telegram
6	Platon Chevychelov	telegram
7	Roman Bokhyan	telegram
8	Rina Zyubanova	telegram
9	Kirill Belyakov	telegram

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 (04.04.2024). 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 (11.04.2024). Classification of cyclic groups. The subgroups of the group of integers. The subgroups of the group Z_n. Left and right cosets, examples. Normal subgroups. The Lagrange theorem and its corollaries.

Lecture 3 (18.04.2024). 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 (25.04.2024). Multiplicative 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 (27.04.2024). 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 (23.05.2024). 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 (30.05.2024). 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 (06.06.2024). 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 (13.06.2024). 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 (17.06.2024). 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. If you send the solution after the deadline your grade will be multiplied by 0.7^t where t -- is the time passed after the deadline in days (not rounded). So, it is not an issue to send your work 1 or 2 hours after the deadline.

Seminar 1 (04.04.2024). Problems Since I am very late with publishing the homework there is a special deadline in groups 232, 233, 235. They should send their solutions till 23:00 April, 14 on Sunday.

Seminar 2 (12.04.2024). Problems Deadline for 233 is April, 21, 23:00. Deadline for 235 is April, 22, 23:00.

Seminar 3 (18.04.2024). Problems

Seminar 4 (25.04.2024). Problems Deadline for 232: May, 10th, 9:30.

Seminar 5 (08.05.2024). Problems Deadline for 233: May, 15th, 23:00; deadline for 232: May, 18th, 9:30.

Seminar 6 (25.05.2024). Problems Deadline for 233 and 235: June, 1th, 23:00;

Seminar 7 (02.06.2024). Problems Deadline for 233 and 235: June, 9th, 23:00;

Seminar 8 (06.06.2024). Problems

Seminar 9 (13.06.2024). Problems

Results

• Homework

231	222	223	234	235

• Test

231	222	223	234	235

• Summary Statement

231	222	223	234	235

Links

• Telegram channel of the course.

• Lecture Notes