Algebra DSBA 2021/2022

Teachers and assistants

Группа	211	212	213	214
Lecturer	Dima Trushin Telegram
Teacher	Dima Trushin	Andrew Mazhuga	Nikita Medved	Galina Kaleeva
Assistant	Oleg Ivanov	Milena Morozova	Kirill Khodakovskiy	Dasha Ivanova

Consultations schedule

	Teacher/Assistant	How to contact	When
1	Dima Trushin	telegram	Write me and we will schedule a meeting
2	Andrew Mazhuga
3	Nikita Medved	telegram	Write me and we will schedule a meeting
4	Galina Kaleeva		Saturday, 13:00, zoom May, 14th, 15.00 Sign up
5	Oleg Ivanov
6	Milena Morozova
7	Kirill Khodakovskiy
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 (08.04.2022). 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. Classification of cyclic groups.

Lecture 2 (15.04.2022). 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 (22.04.2022). 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 (29.04.2022). 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 (13.05.2022). 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 (20.05.2022). 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 (27.05.2022). 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 (03.06.2022). 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 (10.06.2022). 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 (17.06.2022). 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 submit your homework after the deadline you score will be multiplied by 0.7^(t/24), where t is the amount of hours of the delay.

Seminar 1 (08.04.2022). Problems

Seminar 2 (15.04.2022). Problems

Seminar 3 (22.04.2022). Problems

Seminar 4 (29.04.2022). Problems

Since Homework 4 is late, there is a special deadline for Group 211. The new deadline is Saturday 23:00 May 14. Deadlines for the other groups remain the same.

Seminar 5 (13.05.2022). Problems

Seminar 6 (20.05.2022). Problems

Seminar 7 (27.05.2022). Problems

Since Homework 7 is late, there is a special deadline for Group 211. The new deadline is Saturday 23:00 June 4. Deadlines for the other groups remain the same.

Seminar 8 (03.06.2022). Problems

Seminar 9 (10.06.2022). Problems

Deadlines for group 214

Home assignment 3: 16.05.2022, 9.30.

Home assignment 4: 21.05.2022, 9.30.

Home assignment 5: 23.05.2022, 9.30.

Test

• Offline rules
• Online rules

Results

• Homework

211	212	213	214

• Summary Statement

211	212	213	214

Links

• Lecture Notes
• 203 seminars (Medved)