# Algebra DSBA 2020/2021

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# 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

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

# Exam

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

# Results

• Homework
201 202 203 204
• Summary Statement
201 202 203 204