# DSBA Algebra 2022 2023

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# Teachers and assistants

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Lecturer Dima Trushin Telegram
Teacher Dima Trushin Andrew Mazhuga Nikita Medved Galina Kaleeva
Assistant Misha Gorodov Dasha Ivanova Artem Makarenkov

# Consultations schedule

Teacher/Assistant How to contact When
1
Dima Trushin telegram Write me and we will schedule a meeting
2
Andrew Mazhuga telegram
3
Nikita Medved
4
Galina Kaleeva Tuesday, 18:00, zoom. Please notify me beforehand
5
Misha Gorodov telegram
6
Dasha Ivanova telegram
7
Artem Makarenkov telegram
8
Alena Chislova telegram

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 (06.04.2023). 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 (13.04.2023). 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 (20.04.2023). 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 (27.04.2023). 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. RSA.

Lecture 5 (11.05.2023). 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 (18.05.2023). 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 (25.05.2023). 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 (01.06.2023). 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 (08.06.2023). 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 (15.06.2023). The Diamond Lemma. A proof of the Buchberger criterion. The Dickson Lemma and termination of the Buchberger algorithm.

# Problem sheets

The solutions should be sent to your teaching assistant before the beginning of the next seminar. The deadline is strict. We do not evaluate the homework sent after the deadline.

Seminar 1 (06.04.2023). Problems

Seminar 2 (13.04.2023). Problems

Seminar 3 (20.04.2023). Problems

Seminar 4 (27.04.2023). Problems

Seminar 5 (11.05.2023). Problems

Seminar 6 (18.05.2023). Problems

Seminar 7 (25.05.2023). Problems

Seminar 8 (01.06.2023). Problems

Seminar 9 (08.06.2023). Problems

# Test

The test will take place on Monday 19 of June, since 10:00 in online format. The following file contains all the information.

# Exam

The exam will take place on June 24, Saturday.

• List of definitions and statements.
• List of statements to prove.
• The rules for the exam.

The schedule for the exam. You must come at the time in the schedule.

# Results

• Homework
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• Test
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• Summary Statement
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