Algebra DSBA 2021/2022 — различия между версиями
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'''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 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. | ||
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| + | '''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. | ||
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| + | '''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. | ||
= Problem sheets = | = Problem sheets = | ||
Версия 23:09, 14 мая 2022
Содержание
[убрать]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 | |
|---|---|---|---|
| |
Dima Trushin | telegram | Write me and we will schedule a meeting |
| |
Andrew Mazhuga | ||
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Nikita Medved | telegram | Write me and we will schedule a meeting |
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Galina Kaleeva | May, 14th, 15.00 Sign up | |
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Oleg Ivanov | ||
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Milena Morozova | ||
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Kirill Khodakovskiy | ||
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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.
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.
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.
Exam
Results
- Homework
| 211 | 212 | 213 | 214 |
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- Summary Statement
| [211] | [212] | [213] | [214] |
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