Algebra DSBA 2019/2020

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Schedule

Teachers and assistants

Группа 191 192 193
Lecturer Dima Trushin
Teacher Dima Trushin Sergey Gayfullin Galina Kaleeva
Assistant Arina Yunying Timur

Consultations schedule

Teacher/Assistant Monday Tuesday Wednesday Thursday Friday
1
Dima Trushin zoom since 16:00
2
Sergey Gayfullin 16:30–18:00
3
Galina Kaleeva 16:30–18:00 пароль: algebra
4
Arina
5
Yunying
6
Timur

Grading system

The cumulative grade is computed as follows:

C = 0,6 * H + 0,4 * T,

where H is the grade for the home assignments and T is the written test grade.

The final course grade is given by

F = 0,5 * C + 0,5 * E = 0,3 * H + 0,2 T + 0,5 E

where 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 (06.04.2020). 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 (13.04.2020). Subgroups of the group of integers. Left and right cosets, examples. Normal subgroups. The Lagrange theorem and its 5 corollaries.

Lecture 3 (20.04.2020). 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 (27.04.2020). 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 (07.05.2020). 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 (12.05.2020). 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 (18.05.2020). 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.

Problem sheets

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

Seminar 1 (06.04.2020). Problems

Seminar 2 (13.04.2020). Problems

Seminar 3 (20.04.2020). Problems

Seminar 4 (27.04.2020). Problems

Seminar 5 (07.05.2020). Problems

Seminar 6 (12.05.2020). Problems

Seminar 7 (18.05.2020). Problems

Seminar 8 (25.05.2020). Problems

Results

191 192 193

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